Incenter–excenter lemma

In geometry, the incenter–excenter lemma is the theorem that the line segment between the incenter and any excenter of a triangle, or between two excenters, is the diameter of a circle (an incenter–excenter or excenter–excenter circle) also passing through two triangle vertices with its center on the circumcircle.

[1][2][3] This theorem is best known in Russia, where it is called the trillium theorem (теорема трилистника) or trident lemma (лемма о трезубце), based on the geometric figure's resemblance to a trillium flower or trident;[4][5] these names have sometimes also been adopted in English.

[8][2] The theorem is helpful for solving competitive Euclidean geometry problems,[1] and can be used to reconstruct a triangle starting from one vertex, the incenter, and the circumcenter.

Let I be its incenter and let D be the point where line BI (the angle bisector of ∠ABC) crosses the circumcircle of ABC.

Then, the theorem states that D is equidistant from A, C, and I. Equivalently: A fourth point E, the excenter of ABC relative to B, also lies at the same distance from D, diametrically opposite from I.

[13] Other triangle reconstruction problems, such as the reconstruction of a triangle from a vertex, incenter, and center of its nine-point circle, can be solved by reducing the problem to the case of a vertex, incenter, and circumcenter.

The circle with IJ as diameter passes through the other two vertices and is centered on the circumcircle of ABC.