Q:

How to prove that the points $A, Q, M, R$ lie on a circle?

On the sides $BC, CA, AB $ isosceles triangle ABC, where  $AB = BC$, was chosen respectively points $P, Q, R.  BQ$ and $CR$ intersect at a point M. Let  $ \angle BRP = \angle CPQ = \angle BAC$. How to prove that the points $A, Q, M, R$ lie on a circle?  Is there a simple way?

A:

Let's observe some simple things:
$\angle ACB=\angle ARP$, so $BRPC$ is an inscribed quadrilateral and $\angle PBC=\angle CRP$,
Also $\angle ABC=\angle CPQ$ so $BAPQ$ is an inscribed quadrilateral, too and $\angle PBC=\angle PAQ$
Now we could see $\angle PAQ=\angle CRP$, furthermore, $\angle AQB=\angle CAQ+\angle ACQ=\angle CRA$.
So $BRMQ$ is an inscribed quadrilateral. 

