# Prove incenter is orthocenter.

• Feb 3rd 2013, 01:13 PM
TimsBobby2
Prove incenter is orthocenter.
Let Ia, Ib, Ic, be the excenters of triangle ABC. Prove that A, B, and C are the feet of the altitudes of triangle IaIbIc and that the incenter I of triangle ABC is the orthocenter of triangle IaIbIc.

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.
• Feb 3rd 2013, 01:51 PM
ILikeSerena
Re: Prove incenter is orthocenter.
Quote:

Originally Posted by TimsBobby2
Let Ia, Ib, Ic, be the excenters of triangle ABC. Prove that A, B, and C are the feet of the altitudes of triangle IaIbIc and that the incenter I of triangle ABC is the orthocenter of triangle IaIbIc.

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 IaIbA and the lines AB and AC.
The line AIa divides the angle between AB and AC in two equal angles.
The line AIb divides the angle between AC and BA in two equal angles.
This means that the 2 dividing lines AIa and AIb are perpendicular.
In other words, A is the foot of an altitude of triangle IaIbIc.