geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, the incenter–excenter lemma is the

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...

that the line segment between the incenter and any excenter of a triangle, or between two excenters, is the

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...

of a

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 ...

(an ''incenter–excenter'' or ''excenter–excenter circle'') also passing through two triangle vertices with its center on 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 ...

. This theorem is best known in Russia, where it is called the trillium theorem () or trident lemma (), based on the geometric figure's resemblance to a

trillium ''Trillium'' (trillium, wakerobin, toadshade, tri flower, birthroot, birthwort, and sometimes "wood lily") is a genus of about fifty flowering plant species in the family Melanthiaceae. ''Trillium'' species are native to temperate regions of No ...

flower or

trident A trident is a three- pronged spear. It is used for spear fishing and historically as a polearm. The trident is the weapon of Poseidon, or Neptune, the God of the Sea in classical mythology. The trident may occasionally be held by other mari ...

; these names have sometimes also been adopted in English. These relationships arise because the incenter and excenters of any triangle form an

orthocentric system In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through disjoint pairs among the points are perpendicular, and ...

whose

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 ea ...

is the circumcircle of the original triangle. The theorem is helpful for solving competitive Euclidean geometry problems, and can be used to reconstruct a triangle starting from one vertex, the incenter, and the circumcenter.

Statement

Let be an arbitrary

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 ...

. Let be its

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 ...

and let be the point where line (the

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 ...

of ) crosses the

of . Then, the theorem states that is

equidistant A point is said to be equidistant from a set of objects if the distances between that point and each object in the set are equal. In two-dimensional Euclidean geometry, the locus of points equidistant from two given (different) points is the ...

from , , and . Equivalently: *The circle through , , and has its center at . In particular, this implies that the center of this circle lies on the circumcircle. *The three triangles , , and are

isosceles In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

, with as their apex. A fourth point , the

excenter 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 relative to , also lies at the same distance from , diametrically opposite from .

Proof

By the

inscribed angle theorem In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Equivalently, an i ...

\angle IBA = \angle DCA, \ \angle IBC = \angle DAC.

Since

BI

is an angle bisector,

\angle DCA = \angle DAC \implies AD = CD.

We also get :

\begin
\angle DIA &= 180^\circ - \angle AIB \\
&= 180^\circ - (180^\circ - \angle IAB - \angle IBA) \\
&= \angle IAB + \angle IBA \\
&= \angle IAC + \angle DAC \\
&= \angle IAD \\
\implies AD &= DI.
\end

Application to triangle reconstruction

This theorem can be used to reconstruct a triangle starting from the locations only of one vertex, the

, and the

circumcenter 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 ...

of the triangle. For, let be the given vertex, be the incenter, and be the circumcenter. This information allows the successive construction of: *the circumcircle of the given triangle, as the circle with center and radius , *point as the intersection of the circumcircle with line , *the circle of the theorem, with center and radius , and *vertices and as the intersection points of the two circles. However, for some triples of points , , and , this construction may fail, either because line is tangent to the circumcircle or because the two circles do not have two crossing points. It may also produce a triangle for which the given point is an excenter rather than the incenter. In these cases, there can be no triangle having as vertex, as incenter, and as circumcenter. Other triangle reconstruction problems, such as the reconstruction of a triangle from a vertex, incenter, and center of its

, can be solved by reducing the problem to the case of a vertex, incenter, and circumcenter.

Generalization

Let and be any two of the four points given by the incenter and the three excenters of a triangle . Then and are collinear with one of the three triangle vertices. The circle with as diameter passes through the other two vertices and is centered on the circumcircle of . When one of or is the incenter, this is the trillium theorem, with line as the (internal) angle bisector of one of the triangle's angles. However, it is also true when and are both excenters; in this case, line is the external angle bisector of one of the triangle's angles.

References

{{Reflist, 30em, refs = {{cite book , last1 = Aref , first1 = M. N. , last2 = Wernick , first2 = William , year = 1968 , title = Problems and Solutions in Euclidean Geometry , publisher=Dover , isbn = 9780486477206 , at = 3.3(i), {{p., 68, url=https://books.google.com/books?id=vAcU7jOFhG4C&pg=PA68. {{cite web , last = Bogomolny , first = Alexander , author-link = Alexander Bogomolny , title = Midpoints of the Lines Joining In- and Excenters , work = Cut-the-Knot , url = http://www.cut-the-knot.org/Curriculum/Geometry/InExCircum.shtml , access-date = 2016-01-26 {{cite web , last = Bogomolny , first = Alexander , author-link = Alexander Bogomolny , title = A Property of Circle Through the Incenter , work = Cut-the-Knot , url = http://www.cut-the-knot.org/m/Geometry/OnePropertyOfCircleThroughIncenter.shtml , access-date = 2016-01-26 {{cite book , last1 = Chou , first1 = Shang-Ching , last2 = Gao , first2 = Xiao-Shan , last3 = Zhang , first3 = Jingzhong , year = 1994 , title = Machine Proofs in Geometry , publisher = World Scientific , isbn = 9789810215842 , at = Examples 6.145 and 6.146, {{pgs, 328–329 , url = https://books.google.com/books?id=rnI7H_7bLacC&pg=PA328 . {{cite book , last = Chen , first = Evan , year = 2016 , chapter = §1.4 The Incenter/Excenter Lemma , title = Euclidean Geometry in Mathematical Olympiads , publisher = Mathematical Association of America , pages = 9–10 , isbn = 9780883858394 {{cite journal , last1 = Garcia , first1 = Ronaldo , last2 = Odehnal , first2 = Boris , last3 = Reznik , first3 = Dan , year = 2022 , title = Loci of poncelet triangles in the general closure case , journal = Journal of Geometry , volume = 113 , number = 1 , page = 17 , doi = 10.1007/s00022-022-00629-3 , arxiv = 2108.05430 {{cite book , last = Johnson , first = Roger A. , year = 1929 , title = Modern Geometry , publisher = Houghton Mifflin , chapter = X. Inscribed and Escribed Circles , pages = 182–194 , chapter-url = https://archive.org/details/moderngeometry0000unse_q5z5/page/182/ , chapter-url-access = limited Trillium theorem: {{cite web , url = http://www.geometry.ru/persons/kushnir/9pointcircle.pdf , title = Это открытие - золотой ключ Леонарда Эйлера , at = Ф7 (Теорема трилистника), page 34; proof on page 36 , author = И. А. Кушнир , language = ru {{pb Trident lemma: {{cite web , title = Задачи для школьного математического кружка , author1=Р. Н. Карасёв , author2=В. Л. Дольников , author3=И. И. Богданов , author4=А. В. Акопян , pages = 4 , location = Problem 1.2 , url = http://www.rkarasev.ru/common/upload/taskprob.pdf , language = ru {{cite journal , last1 = Le , first1 = Nguyen , last2 = Wildberger , first2 = Norman , year = 2016 , title = Incenter Symmetry, Euler Lines, and Schiffler Points , journal = KoG , volume = 20 , number = 20 , pages = 22–30 , url = https://hrcak.srce.hr/174092 {{citation , last = Morris , first = Richard , issue = 2 , journal =

The Mathematics Teacher Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds an ...

, jstor = 27951001 , pages = 63–71 , title = Circles through notable points of the triangle , volume = 21 , year = 1928, doi = 10.5951/MT.21.2.0069 . See in particular the discussion on p. 65 of circles {{mvar, BIC, {{mvar, CIA, {{mvar, AIB, and their centers. {{cite web , url = http://math.mosolymp.ru/upload/files/2015/aesc/approaching-2014-10-29-inscribed-angles.pdf , publisher = СУНЦ МГУ им. М. В. Ломоносова - школа им. А.Н. Колмогорова , title = 6. Лемма о трезубце , date = 2014-10-29 , language = ru {{cite book , last = Weisstein , first = Eric W. , year = 1999 , title = CRC Concise Encyclopedia of Mathematics , publisher = CRC Press , isbn = 0849396409 , at
"Excenter–Excenter Circle"
p. 591
"Incenter–Excenter Circle"
p. 894 Republished at ''

MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ...

''
"Excenter–Excenter Circle"
{{citation , last = Yiu , first = Paul , issue = 2 , journal = Journal for Geometry and Graphics , mr = 3088369 , pages = 171–183 , title = Conic construction of a triangle from its incenter, nine-point center, and a vertex , url = http://math.fau.edu/Yiu/j16h2yiu.pdf , volume = 16 , year = 2012 {{cite book , last1 = Zaslavsky , first1 = Alexey A. , last2 = Skopenkov , first2 = Mikhail B. , year = 2021 , title = Mathematics via Problems. Part 2: Geometry , publisher = American Mathematical Society , isbn = 9781470448790 , page = 15 Theorems about triangles and circles

Statement

Proof

Application to triangle reconstruction

Generalization

See also

References