geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the tangential triangle of a reference

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

(other than a

right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...

) is the triangle whose sides are on the

tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

s to the reference triangle's

circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

at the reference triangle's vertices. Thus 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 tangential triangle coincides with the circumcircle of the reference triangle. The

circumcenter In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcen ...

of the tangential triangle is on the reference triangle's

Euler line In geometry, the Euler line, named after Leonhard Euler ( ), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, incl ...

, as is the

center of similitude In geometry, a homothetic center (also called a center of similarity or a center of similitude) is a point from which at least two geometrically similar figures can be seen as a dilation or contraction of one another. If the center is externa ...

of the tangential triangle and the

orthic triangle The orthocenter of a triangle, usually denoted by , is the point where the three (possibly extended) altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute. For a right triangle, the orthocenter coi ...

(whose vertices are at the feet of the

altitudes Altitude 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 (e.g., aviation, geometry, geographical s ...

of the reference triangle).Smith, Geoff, and Leversha, Gerry, "Euler and triangle geometry", ''Mathematical Gazette'' 91, November 2007, 436–452. The tangential triangle is homothetic to the

.Altshiller-Court, Nathan. ''College Geometry'', Dover Publications, 2007 (orig. 1952). A reference triangle and its tangential triangle are in perspective, and the axis of perspectivity is the

Lemoine axis In geometry, central lines are certain special straight lines that lie in the plane of a triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This s ...

of the reference triangle. That is, the lines connecting the vertices of the tangential triangle and the corresponding vertices of the reference triangle 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 ...

. The center of perspectivity, where these three lines meet, is the

symmedian point In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the co ...

of the triangle. The tangent lines containing the sides of the tangential triangle are called the exsymmedians of the reference triangle. Any two of these are concurrent with the third

symmedian In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the co ...

of the reference triangle.Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publications, 2007 (orig. 1929). The reference triangle's circumcircle, its

nine-point circle In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: * The midpoint of each s ...

, its

polar circle A polar circle is a geographic term for a conditional circular line (arc) referring either to the Arctic Circle or the Antarctic Circle. These are two of the keynote circles of latitude (parallels). On Earth, the Arctic Circle is currentl ...

, and the circumcircle of the tangential triangle are coaxal. A right triangle has no tangential triangle, because the tangent lines to its circumcircle at its acute vertices are parallel and thus cannot form the sides of a triangle. The reference triangle is the

Gergonne triangle 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 tangential triangle.

References

Objects defined for a triangle {{elementary-geometry-stub

See also

References