Could anyone solve this one for me? I tried to use some kind of variation on Poncelet's theorem, but couldn't find a way to do it. Here it goes:
Let $ABC$ be a triangle and $D\in BC$ such that $AD\perp BC$. Also, let $I$ and $J$ be the incenters of triangles $ABD$ and $ACD$, respectively. The incircles of $ABD$ and $ACD$ are tangents to $AD$ on dots $M$ and $N$, respectively. Finally, let $P$ be the point of tangency of $ABC$'s incircle with side $AB$. The circle centered in $A$ and with radius $AP$ intersects $AD$ in $K$.
a) Show that triangles $IMK$ and $KNJ$ are congruent.
b) Show that $IDJK$ is cyclical.
I didn't have a problem with the image. Here's how I saw it:
I couldn't find a relation between the last circle built and the other ones. And to prove congruence I have to do so, since $K$ is one of both triangles' vertices, and it's built from that specifical circle. So, anyone, please?