excentral

Excentral refers to the excentral triangle associated with a given triangle ABC. The excentral triangle is formed by the three excenters I_A, I_B, and I_C, which are the centers of the A-, B-, and C-excircles (the excircles opposite the corresponding vertices). Each excenter is the intersection point of the internal angle bisector of one angle with the external angle bisectors of the other two angles.

Key properties

- The excentral triangle I_A I_B I_C lies with vertices at the excenters, which are each located on the external angle bisectors of ABC.

- The incenter I of triangle ABC is the orthocenter of the excentral triangle: the lines I I_A, I I_B, and I I_C are perpendicular to the opposite sides I_B I_C, I_C I_A, and I_A I_B, respectively.

- The angles of the excentral triangle relate to the original triangle’s angles by ∠I_B I_A I_C = 90° − A/2, and cyclically.

- The circumcenter of the excentral triangle is known as the Bevan point J. It is the reflection of the incenter I across the circumcenter O of ABC, and it serves as the center of the excentral triangle’s circumcircle through I_A, I_B, and I_C.

- The excentral triangle is acute.

Applications

- The excentral construction is used in triangle geometry to study incenters, excenters, and relations among angle bisectors. It provides a framework for proving identities and for constructing centers such as the Bevan point.

Variants

- In some contexts, the term excentral may refer to objects derived from the excenters or to other constructions that involve excentral relationships tied to a given triangle.