Circle $\Gamma$ is the incircle of $\triangle ABC$ and is also the circumcircle of $\triangle XYZ$. The point $X$ is on $\overline{BC}$, point $Y$ is on $\overline{AB}$, and the point $Z$ is on $\overline{AC}$. If $\angle A=40^\circ$, $\angle B=60^\circ$, and $\angle C=80^\circ$, what is the measure of $\angle AYX$?
+118690 Here is the diagram, and I have introduced a new point, E
BX and BY are both tangents to the circle, Hence they are the same length.
So triangle EYB and triangle EXB are congruent.
So angle YEB = 90 degrees.
So angle EYB = 180-90-30 = 60 degrees
So angle AYX = the supplement of 60 degrees, 180-60 = 120 degrees
