bisectors of exterior angles in triangle

Question

In triangle ABC, the bisectors of the exterior angles B and C meet at H. Show that AH is the bisector of the angle BAC.

I was trying to look at some exterior angles in some triangles but I can't find a way to link those two angles that supposed to be equal. thx!

Do you mean AH, instead of AP? – Arnaldo Sep 23 '22 at 12:21 — Arnaldo, Sep 23 '22 at 12:21
yes, thx. I don't know why I was using P :))) – Numbers Sep 23 '22 at 12:28 — Numbers, Sep 23 '22 at 12:28

score 0 · Answer 1 · answered Sep 23 '22 at 13:23

0

we take the triangle to be made out of lines $l_1,l_2,l_3$ with positive direction into the triangle.
the external bisector of $l_1,l_2$ is the space of points with directed distance $d(P,l_1)=-d(P,l_2)$
and if $P$ is on the external bisector of $l_2,l_3$ then $d(P,l_2)=-d(P,l_3)$
which means $d(P,l_1)=d(P,l_3)$ which means it's on the internal bisector of $l_1,l_3$.

answered Sep 23 '22 at 13:23

razivo

2,205

well, nice one. but I want to solve it using just euclidean geometry. I am sorry for my english – Numbers Sep 23 '22 at 13:39

score 0 · Answer 2 · answered Sep 23 '22 at 19:46

This is a well-know property of triangle excircles. All points that are equidistant from the sides of external angle $B$ lie on its bisector. Ditto for angle $C$. The intersection of two bisectors will be a center of excircle which is tangent to the sides of the external angles which are $BC$ and extensions of $AC$ and $AB$. We can see now that from point $A$ there are two tangents to the circle thus $AH$ will bisect $\angle BAC$ (draw two radii to the points of tangency and you have two congruent right triangles). You can read more about excircles and their properties here

bisectors of exterior angles in triangle

2 Answers2