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.Template:R This theorem is best known in Russia, where it is called the trillium theorem (Template:Lang) or trident lemma (Template:Lang), based on the geometric figure's resemblance to a trillium flower or trident;Template:R these names have sometimes also been adopted in English.Template:R

These relationships arise because the incenter and excenters of any triangle form an orthocentric system whose nine-point circle is the circumcircle of the original triangle.Template:R The theorem is helpful for solving competitive Euclidean geometry problems,Template:R and can be used to reconstruct a triangle starting from one vertex, the incenter, and the circumcenter.

Statement

incenter–excenter lemma with incenter Template:Mvar and excenter Template:Mvar

Let Template:Mvar be an arbitrary triangle. Let Template:Mvar be its incenter and let Template:Mvar be the point where line Template:Mvar (the angle bisector of Template:Math) crosses the circumcircle of Template:Mvar. Then, the theorem states that Template:Mvar is equidistant from Template:Mvar, Template:Mvar, and Template:Mvar. Equivalently:

• The circle through Template:Mvar, Template:Mvar, and Template:Mvar has its center at Template:Mvar. In particular, this implies that the center of this circle lies on the circumcircle.Template:R
• The three triangles Template:Mvar, Template:Mvar, and Template:Mvar are isosceles, with Template:Mvar as their apex.

A fourth point Template:Mvar, the excenter of Template:Mvar relative to Template:Mvar, also lies at the same distance from Template:Mvar, diametrically opposite from Template:Mvar.Template:R

Proof

By the inscribed angle theorem, $\mathrm{\angle }IBA=\mathrm{\angle }DCA,\mathrm{\angle }IBC=\mathrm{\angle }DAC.$

Since $BI$ is an angle bisector, $\mathrm{\angle }DCA=\mathrm{\angle }DAC⟹AD=CD.$

We also get

$\begin{array}{cc}\mathrm{\angle }DIA& =180^{\circ }-\mathrm{\angle }AIB\\ & =180^{\circ }-(180^{\circ }-\mathrm{\angle }IAB-\mathrm{\angle }IBA)\\ & =\mathrm{\angle }IAB+\mathrm{\angle }IBA\\ & =\mathrm{\angle }IAC+\mathrm{\angle }DAC\\ & =\mathrm{\angle }IAD\\ ⟹AD& =DI.\end{array}$

Application to triangle reconstruction

This theorem can be used to reconstruct a triangle starting from the locations only of one vertex, the incenter, and the circumcenter of the triangle. For, let Template:Mvar be the given vertex, Template:Mvar be the incenter, and Template:Mvar be the circumcenter. This information allows the successive construction of:

• the circumcircle of the given triangle, as the circle with center Template:Mvar and radius Template:Mvar,
• point Template:Mvar as the intersection of the circumcircle with line Template:Mvar,
• the circle of the theorem, with center Template:Mvar and radius Template:Mvar, and
• vertices Template:Mvar and Template:Mvar as the intersection points of the two circles.Template:R

However, for some triples of points Template:Mvar, Template:Mvar, and Template:Mvar, this construction may fail, either because line Template:Mvar 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 Template:Mvar is an excenter rather than the incenter. In these cases, there can be no triangle having Template:Mvar as vertex, Template:Mvar as incenter, and Template:Mvar as circumcenter.

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.

Generalization

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

See also

• Angle bisector theorem