In
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
whose sides all can be
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to a single
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
within the quadrilateral. This circle is called the
incircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
of the quadrilateral or its inscribed circle, its center is the ''incenter'' and its radius is called the ''inradius''. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called ''circumscribable quadrilaterals'', ''circumscribing quadrilaterals'', and ''circumscriptible quadrilaterals''.
[ Tangential quadrilaterals are a special case of ]tangential polygon
In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an ''incircle''). This is a circle that is tangent to each of the polygon's sides. The dual pol ...
s.
Other less frequently used names for this class of quadrilaterals are ''inscriptable quadrilateral'', ''inscriptible quadrilateral'', ''inscribable quadrilateral'', ''circumcyclic quadrilateral'', and ''co-cyclic quadrilateral''.[.] Due to the risk of confusion with a quadrilateral that has a circumcircle, which is called a cyclic quadrilateral
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
or inscribed quadrilateral, it is preferable not to use any of the last five names.[
All ]triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
s can have an incircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be tangential is a non-square rectangle. The section characterizations below states what necessary and sufficient condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
s a quadrilateral must satisfy to be able to have an incircle.
Special cases
Examples of tangential quadrilaterals are the kites
A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the face ...
, which include the rhombi
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
, which in turn include the square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
s. The kites are exactly the tangential quadrilaterals that are also orthodiagonal.[.] A right kite
In Euclidean geometry, a right kite is a kite (a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other) that can be inscribed in a circle.Michael de Villiers, ''Some Adventures in Eucl ...
is a kite with a circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
. If a quadrilateral is both tangential and cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in s ...
, it is called a bicentric quadrilateral
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called ''inradius'' and ''circumradius'', and ''incenter'' and ''circumcenter'' r ...
, and if it is both tangential and a trapezoid
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium ().
A trapezoid is necessarily a convex quadrilateral in Eu ...
, it is called a tangential trapezoid
In Euclidean geometry, a tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the incircle or ''inscribed circle''. It is the special case of a tangential ...
.
Characterizations
In a tangential quadrilateral, the four angle bisector
In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
s meet at the center of the incircle. Conversely, a convex quadrilateral in which the four angle bisectors meet at a point must be tangential and the common point is the incenter.[
According to the Pitot theorem, the two pairs of opposite sides in a tangential quadrilateral add up to the same total length, which equals the ]semiperimeter
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate na ...
''s'' of the quadrilateral:
:
Conversely a convex quadrilateral in which ''a'' + ''c'' = ''b'' + ''d'' must be tangential.[
If opposite sides in a convex quadrilateral ''ABCD'' (that is not a ]trapezoid
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium ().
A trapezoid is necessarily a convex quadrilateral in Eu ...
) intersect at ''E'' and ''F'', then it is tangential if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
either of[.]
:
or
:
The second of these is almost the same as one of the equalities in Urquhart's theorem. The only differences are the signs on both sides; in Urquhart's theorem there are sums instead of differences.
Another necessary and sufficient condition is that a convex quadrilateral ''ABCD'' is tangential if and only if the incircles in the two triangles ''ABC'' and ''ADC'' are tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to each other.[.]
A characterization regarding the angles formed by diagonal ''BD'' and the four sides of a quadrilateral ''ABCD'' is due to Iosifescu. He proved in 1954 that a convex quadrilateral has an incircle if and only if[
.]
:
Further, a convex quadrilateral with successive sides ''a'', ''b'', ''c'', ''d'' is tangential if and only if
:
where ''R''''a'', ''R''''b'', ''R''''c'', ''R''''d'' are the radii in the circles externally tangent to the sides ''a'', ''b'', ''c'', ''d'' respectively and the extensions of the adjacent two sides for each side.
Several more characterizations are known in the four subtriangles formed by the diagonals.
Contact points and tangent lengths
The incircle is tangent to each side at one ''point of contact''. These four points define a new quadrilateral inside of the initial quadrilateral: the ''contact quadrilateral,'' which is cyclic as it is inscribed in the initial quadrilateral's incircle.
The eight ''tangent lengths'' (''e'', ''f'', ''g'', ''h'' in the figure to the right) of a tangential quadrilateral are the line segments from a vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
* Vertex (computer graphics), a data structure that describes the positio ...
to the points of contact. From each vertex, there are two congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
tangent lengths.
The two ''tangency chords'' (''k'' and ''l'' in the figure) of a tangential quadrilateral are the line segments that connect contact points on opposite sides. These are also the diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...
s of the contact quadrilateral.
Area
Non-trigonometric formulas
The area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
''K'' of a tangential quadrilateral is given by
:
where ''s'' is the semiperimeter
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate na ...
and ''r'' is the inradius
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
. Another formula is[
:
which gives the area in terms of the diagonals ''p'', ''q'' and the sides ''a'', ''b'', ''c'', ''d'' of the tangential quadrilateral.
The area can also be expressed in terms of just the four tangent lengths. If these are ''e'', ''f'', ''g'', ''h'', then the tangential quadrilateral has the area][
:
Furthermore, the area of a tangential quadrilateral can be expressed in terms of the sides ''a, b, c, d'' and the successive tangent lengths ''e, f, g, h'' as][
:
Since ''eg'' = ''fh'' if and only if the tangential quadrilateral is also cyclic and hence bicentric,][ this shows that the maximal area occurs if and only if the tangential quadrilateral is bicentric.
]
Trigonometric formulas
A trigonometric
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
formula for the area in terms of the sides ''a'', ''b'', ''c'', ''d'' and two opposite angles is[
:
For given side lengths, the area is ]maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
when the quadrilateral is also cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in s ...
and hence a bicentric quadrilateral
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called ''inradius'' and ''circumradius'', and ''incenter'' and ''circumcenter'' r ...
. Then since opposite angles are supplementary angles
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
. This can be proved in another way using calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
.
Another formula for the area of a tangential quadrilateral ''ABCD'' that involves two opposite angles is[
:
where ''I'' is the incenter.
In fact, the area can be expressed in terms of just two adjacent sides and two opposite angles as][
:
Still another area formula is][.]
:
where ''θ'' is either of the angles between the diagonals. This formula cannot be used when the tangential quadrilateral is a kite, since then ''θ'' is 90° and the tangent function is not defined.
Inequalities
As indirectly noted above, the area of a tangential quadrilateral with sides ''a'', ''b'', ''c'', ''d'' satisfies
:
with equality if and only if it is a bicentric quadrilateral
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called ''inradius'' and ''circumradius'', and ''incenter'' and ''circumcenter'' r ...
.
According to T. A. Ivanova (in 1976), the semiperimeter ''s'' of a tangential quadrilateral satisfies
:
where ''r'' is the inradius. There is equality if and only if the quadrilateral is a square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
. This means that for the area ''K'' = ''rs'', there is the inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
:
with equality if and only if the tangential quadrilateral is a square.
Partition properties
The four line segments between the center of the incircle and the points where it is tangent to the quadrilateral partition the quadrilateral into four right kite
In Euclidean geometry, a right kite is a kite (a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other) that can be inscribed in a circle.Michael de Villiers, ''Some Adventures in Eucl ...
s.
If a line cuts a tangential quadrilateral into two polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
s with equal area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
s and equal perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
s, then that line passes through the incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bis ...
.[
]
Inradius
The inradius in a tangential quadrilateral with consecutive sides ''a'', ''b'', ''c'', ''d'' is given by[
:
where ''K'' is the area of the quadrilateral and ''s'' is its semiperimeter. For a tangential quadrilateral with given sides, the inradius is ]maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
when the quadrilateral is also cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in s ...
(and hence a bicentric quadrilateral
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called ''inradius'' and ''circumradius'', and ''incenter'' and ''circumcenter'' r ...
).
In terms of the tangent lengths, the incircle has radius[.]
:
The inradius can also be expressed in terms of the distances from the incenter ''I'' to the vertices of the tangential quadrilateral ''ABCD''. If ''u = AI'', ''v = BI'', ''x = CI'' and ''y = DI'', then
:
where .
If the incircles in triangles ''ABC'', ''BCD'', ''CDA'', ''DAB'' have radii respectively, then the inradius of a tangential quadrilateral ''ABCD'' is given by
:
where .
Angle formulas
If ''e'', ''f'', ''g'' and ''h'' are the tangent lengths from the vertices ''A'', ''B'', ''C'' and ''D'' respectively to the points where the incircle is tangent to the sides of a tangential quadrilateral ''ABCD'', then the angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
s of the quadrilateral can be calculated from[
:
:
:
:
The angle between the tangency chords ''k'' and ''l'' is given by][
:
]
Diagonals
If ''e'', ''f'', ''g'' and ''h'' are the tangent lengths from ''A'', ''B'', ''C'' and ''D'' respectively to the points where the incircle is tangent to the sides of a tangential quadrilateral ''ABCD'', then the lengths of the diagonals ''p = AC'' and ''q = BD'' are[
:
:
]
Tangency chords
If ''e'', ''f'', ''g'' and ''h'' are the tangent lengths of a tangential quadrilateral, then the lengths of the tangency chords are[
:
:
where the tangency chord of length ''k'' connects the sides of lengths ''a'' = ''e'' + ''f'' and ''c'' = ''g'' + ''h'', and the one of length ''l'' connects the sides of lengths ''b'' = ''f'' + ''g'' and ''d'' = ''h'' + ''e''. The squared ratio of the tangency chords satisfies][
:
The two tangency chords
*are ]perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...
if and only if the tangential quadrilateral also has a circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
(it is bicentric).[
*have equal lengths if and only if the tangential quadrilateral is a ]kite
A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the fac ...
.[
The tangency chord between the sides ''AB'' and ''CD'' in a tangential quadrilateral ''ABCD'' is longer than the one between the sides ''BC'' and ''DA'' if and only if the ]bimedian
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
between the sides ''AB'' and ''CD'' is shorter than the one between the sides ''BC'' and ''DA''.
If tangential quadrilateral ''ABCD'' has tangency points ''W'' on ''AB'' and ''Y'' on ''CD'', and if tangency chord ''WY'' intersects diagonal ''BD'' at ''M'', then the ratio of tangent lengths equals the ratio of the segments of diagonal ''BD''.
Collinear points
If ''M1'' and ''M2'' are the midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
Formula
The midpoint of a segment in ''n''-dimen ...
s of the diagonals ''AC'' and ''BD'' respectively in a tangential quadrilateral ''ABCD'' with incenter ''I'', and if the pairs of opposite sides meet at ''J'' and ''K'' with ''M3'' being the midpoint of ''JK'', then the points ''M3'', ''M1'', ''I'', and ''M2'' are collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
.[ The line containing them is the ]Newton line
In Euclidean geometry the Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral with at most two parallel sides.Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. ...
of the quadrilateral.
If the extensions of opposite sides in a tangential quadrilateral intersect at ''J'' and ''K'', and the extensions of opposite sides in its contact quadrilateral intersect at ''L'' and ''M'', then the four points ''J'', ''L'', ''K'' and ''M'' are collinear.[.]
If the incircle is tangent to the sides ''AB'', ''BC'', ''CD'', ''DA'' at ''T1'', ''T2'', ''T3'', ''T4'' respectively, and if ''N1'', ''N2'', ''N3'', ''N4'' are the isotomic conjugates of these points with respect to the corresponding sides (that is, ''AT1'' = ''BN1'' and so on), then the ''Nagel point'' of the tangential quadrilateral is defined as the intersection of the lines ''N1N3'' and ''N2N4''. Both of these lines divide the perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
of the quadrilateral into two equal parts. More importantly, the Nagel point ''N'', the "area centroid" ''G'', and the incenter ''I'' are collinear in this order, and ''NG'' = 2''GI''. This line is called the ''Nagel line'' of a tangential quadrilateral.[.]
In a tangential quadrilateral ''ABCD'' with incenter ''I'' and where the diagonals intersect at ''P'', let ''HX'', ''HY'', ''HZ'', ''HW'' be the orthocenter
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...
s of triangles ''AIB'', ''BIC'', ''CID'', ''DIA''. Then the points ''P'', ''HX'', ''HY'', ''HZ'', ''HW'' are collinear.[
]
Concurrent and perpendicular lines
The two diagonals and the two tangency chords are concurrent
Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to:
Law
* Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea''
* Concurring opinion (also called a "concurrence"), a ...
.[Yiu, Paul, ''Euclidean Geometry'']
1998, pp. 156–157.[Grinberg, Darij, ''Circumscribed quadrilaterals revisited'', 2008]
/ref> One way to see this is as a limiting case of Brianchon's theorem, which states that a hexagon all of whose sides are tangent to a single conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
has three diagonals that meet at a point. From a tangential quadrilateral, one can form a hexagon with two 180° angles, by placing two new vertices at two opposite points of tangency; all six of the sides of this hexagon lie on lines tangent to the inscribed circle, so its diagonals meet at a point. But two of these diagonals are the same as the diagonals of the tangential quadrilateral, and the third diagonal of the hexagon is the line through two opposite points of tangency. Repeating this same argument with the other two points of tangency completes the proof of the result.
If the extensions of opposite sides in a tangential quadrilateral intersect at ''J'' and ''K'', and the diagonals intersect at ''P'', then ''JK'' is perpendicular to the extension of ''IP'' where ''I'' is the incenter.[
]
Incenter
The incenter of a tangential quadrilateral lies on its Newton line
In Euclidean geometry the Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral with at most two parallel sides.Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. ...
(which connects the midpoints of the diagonals).
The ratio of two opposite sides in a tangential quadrilateral can be expressed in terms of the distances between the incenter ''I'' and the vertices according to[
:
The product of two adjacent sides in a tangential quadrilateral ''ABCD'' with incenter ''I'' satisfies
:
If ''I'' is the incenter of a tangential quadrilateral ''ABCD'', then][
:
The incenter ''I'' in a tangential quadrilateral ''ABCD'' coincides with the "vertex centroid" of the quadrilateral ]if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
[
:
If ''Mp'' and ''Mq'' are the ]midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
Formula
The midpoint of a segment in ''n''-dimen ...
s of the diagonals ''AC'' and ''BD'' respectively in a tangential quadrilateral ''ABCD'' with incenter ''I'', then [
:
where ''e'', ''f'', ''g'' and ''h'' are the tangent lengths at ''A'', ''B'', ''C'' and ''D'' respectively. Combining the first equality with a previous property, the "vertex centroid" of the tangential quadrilateral coincides with the incenter if and only if the incenter is the midpoint of the line segment connecting the midpoints of the diagonals.
If a ]four-bar linkage
In the study of mechanisms, a four-bar linkage, also called a four-bar, is the simplest closed-chain movable linkage. It consists of four bodies, called ''bars'' or ''links'', connected in a loop by four joints. Generally, the joints are config ...
is made in the form of a tangential quadrilateral, then it will remain tangential no matter how the linkage is flexed, provided the quadrilateral remains convex. (Thus, for example, if a square is deformed into a rhombus it remains tangential, though to a smaller incircle). If one side is held in a fixed position, then as the quadrilateral is flexed, the incenter traces out a circle of radius where ''a,b,c,d'' are the sides in sequence and ''s'' is the semiperimeter.
Characterizations in the four subtriangles
In the nonoverlapping triangles ''APB'', ''BPC'', ''CPD'', ''DPA'' formed by the diagonals in a convex quadrilateral ''ABCD'', where the diagonals intersect at ''P'', there are the following characterizations of tangential quadrilaterals.
Let ''r''1, ''r''2, ''r''3, and ''r''4 denote the radii of the incircles in the four triangles ''APB'', ''BPC'', ''CPD'', and ''DPA'' respectively. Chao and Simeonov proved that the quadrilateral is tangential if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
:
This characterization had already been proved five years earlier by Vaynshtejn.
In the solution to his problem, a similar characterization was given by Vasilyev and Senderov. If ''h''1, ''h''2, ''h''3, and ''h''4 denote the altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
s in the same four triangles (from the diagonal intersection to the sides of the quadrilateral), then the quadrilateral is tangential if and only if[
:
Another similar characterization concerns the exradii ''r''''a'', ''r''''b'', ''r''''c'', and ''r''''d'' in the same four triangles (the four ]excircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
s are each tangent to one side of the quadrilateral and the extensions of its diagonals). A quadrilateral is tangential if and only if[
:
If ''R''1, ''R''2, ''R''3, and ''R''4 denote the radii in the ]circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
s of triangles ''APB'', ''BPC'', ''CPD'', and ''DPA'' respectively, then the quadrilateral ''ABCD'' is tangential if and only if
:
In 1996, Vaynshtejn was probably the first to prove another beautiful characterization of tangential quadrilaterals, that has later appeared in several magazines and websites.[ It states that when a convex quadrilateral is divided into four nonoverlapping triangles by its two diagonals, then the incenters of the four triangles are concyclic if and only if the quadrilateral is tangential. In fact, the incenters form an orthodiagonal cyclic quadrilateral.][ A related result is that the incircles can be exchanged for the excircles to the same triangles (tangent to the sides of the quadrilateral and the extensions of its diagonals). Thus a convex quadrilateral is tangential if and only if the excenters in these four ]excircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
s are the vertices of a cyclic quadrilateral
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
.[
A convex quadrilateral ''ABCD'', with diagonals intersecting at ''P'', is tangential if and only if the four excenters in triangles ''APB'', ''BPC'', ''CPD'', and ''DPA'' opposite the vertices ''B'' and ''D'' are concyclic.][ If ''Ra'', ''Rb'', ''Rc'', and ''Rd'' are the exradii in the triangles ''APB'', ''BPC'', ''CPD'', and ''DPA'' respectively opposite the vertices ''B'' and ''D'', then another condition is that the quadrilateral is tangential if and only if][
:
Further, a convex quadrilateral ''ABCD'' with diagonals intersecting at ''P'' is tangential if and only if][
:
where ∆(''APB'') is the area of triangle ''APB''.
Denote the segments that the diagonal intersection ''P'' divides diagonal ''AC'' into as ''AP'' = ''p''1 and ''PC'' = ''p''2, and similarly ''P'' divides diagonal ''BD'' into segments ''BP'' = ''q''1 and ''PD'' = ''q''2. Then the quadrilateral is tangential if and only if any one of the following equalities are true:
:
or][
:
or][
:
]
Conditions for a tangential quadrilateral to be another type of quadrilateral
Rhombus
A tangential quadrilateral is a rhombus
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
if and only if its opposite angles are equal.[.]
Kite
A tangential quadrilateral is a kite
A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the fac ...
if and only if any one of the following conditions is true:[.]
*The area is one half the product of the diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...
s.
*The diagonals are perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...
.
*The two line segments connecting opposite points of tangency have equal lengths.
*One pair of opposite tangent lengths have equal lengths.
*The bimedians have equal lengths.
*The products of opposite sides are equal.
*The center of the incircle lies on the diagonal that is the axis of symmetry.
Bicentric quadrilateral
If the incircle is tangent to the sides ''AB'', ''BC'', ''CD'', ''DA'' at ''W'', ''X'', ''Y'', ''Z'' respectively, then a tangential quadrilateral ''ABCD'' is also cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in s ...
(and hence bicentric) if and only if any one of the following conditions hold:[
*''WY'' is perpendicular to ''XZ''
*
*
The first of these three means that the ''contact quadrilateral'' ''WXYZ'' is an ]orthodiagonal quadrilateral
In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular ...
.
A tangential quadrilateral is bicentric if and only if its inradius is greater than that of any other tangential quadrilateral having the same sequence of side lengths.[.]
Tangential trapezoid
If the incircle is tangent to the sides ''AB'' and ''CD'' at ''W'' and ''Y'' respectively, then a tangential quadrilateral ''ABCD'' is also a trapezoid
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium ().
A trapezoid is necessarily a convex quadrilateral in Eu ...
with parallel sides ''AB'' and ''CD'' if and only if[.]
:
and ''AD'' and ''BC'' are the parallel sides of a trapezoid if and only if
:
See also
*Circumscribed circle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
*Ex-tangential quadrilateral
In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the ''extensions'' of all four sides are tangent to a circle outside the quadrilateral.Radic, Mirko; Kaliman, Zoran and Kadum, Vladimir, "A condition that a tan ...
*Tangential triangle
In geometry, the tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at the reference triangle's vertices. Thus the incircle of the ...
*Tangential polygon
In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an ''incircle''). This is a circle that is tangent to each of the polygon's sides. The dual pol ...
References
External links
*
{{Polygons
Types of quadrilaterals