Thread: Prove incenter is orthocenter.

Let I_a, I_b, I_c, be the excenters of triangle ABC. Prove that A, B, and C are the feet of the altitudes of triangle I_aI_bI_c and that the incenter I of triangle ABC is the orthocenter of triangle I_aI_bI_c.


It seems obvious to me that once you show that A, B, and C are the feets of the altitudes of the triangle joining the excenters of triangle ABC that the incenter I of triangle ABC is the orthocenter of the other triangle because they are concurrent inside triangle ABC. However, how would I show that A, B, and C are the feet of the altitudes? Any help would be appreciated.

Originally Posted by TimsBobby2

Let I_a, I_b, I_c, be the excenters of triangle ABC. Prove that A, B, and C are the feet of the altitudes of triangle I_aI_bI_c and that the incenter I of triangle ABC is the orthocenter of triangle I_aI_bI_c.


It seems obvious to me that once you show that A, B, and C are the feets of the altitudes of the triangle joining the excenters of triangle ABC that the incenter I of triangle ABC is the orthocenter of the other triangle because they are concurrent inside triangle ABC. However, how would I show that A, B, and C are the feet of the altitudes? Any help would be appreciated.

Hi TimsBobby2!

Suppose we take a look at the triangle I_aI_bA and the lines AB and AC.
The line AI_a divides the angle between AB and AC in two equal angles.
The line AI_b divides the angle between AC and BA in two equal angles.
This means that the 2 dividing lines AI_a and AI_b are perpendicular.
In other words, A is the foot of an altitude of triangle I_aI_bI_c.