Question

In a triangle ΔABC, Let O be the circumcenter, and let I be the incenter. Let the line AI intersect the circumcircle at D. Prove ΔBID is isosceles.

Answer 1

I assume that I just don't know enough about all the centers of a triangle and the related theorems to see the result to this. Either way, don't know what connections to make. All I was able to do was find that a couple of the triangles are similar, but not sure that gets me anywhere. I would assume that the circumcenter has to be involved somehow, but I'm not sure how.

Answer 2

Right, what you have drawn looks good, similar to what I had drawn up. From there, wasn't sure where to go. All I could do was determine that certain triangles were similar, otherwise no idea.

Answer 3

I also don't know where to go from here.

Answer 4

No worries :) Anyone is at least willing to take a look or consider a solution is appreciated

Answer 5

Thanks.

Answer 6

@ganeshie8

Answer 7

The center I is the center of the incircle. To create the incircle based on the triangle, we find the angle bisectors of triangle ABC

so these two angles are congruent (because IB bisects angle ABC)



angle ABI = angle CBI