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_{a}I_{b}I_{c} and that the incenter I of triangle ABC is the orthocenter of triangle I_{a}I_{b}I_{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.

Re: Prove incenter is orthocenter.

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**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_{a}I_{b}I_{c} and that the incenter I of triangle ABC is the orthocenter of triangle I_{a}I_{b}I_{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_{a}I_{b}A 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_{a}I_{b}I_{c}.