Circle Gamma is the incircle ABC of and is also the circumcircle of XYZ. The point X is on BC, point Y is on AB, and the point is Z on AC. If angle A = 40, angle B = 80, and angle C = 60, what is the measure of angle AYX?
So, this is the diagram of the question:
The point I is the center of the incircle, i.e., the incenter of triangle ABC.
By tangent properties, we have \(\angle IYB = \angle IXB = 90^\circ\).
Since \(\angle IYB + \angle IXB = 180^\circ\), the quadrilateral IYBX is a cyclic quadrilateral, so \(\angle XIY = 180^\circ - \angle XBY = 180^\circ - 80^\circ = 100^\circ\).
Now, since \(IX\) and \(IY\) are both radii of a circle, \(\triangle IXY\) is isosceles. Using this information, we can find the size of \(\angle IYX\). Then just note that \(\angle AYI = 90^\circ\), and add that to \(\angle IYX\). You will get \(\angle AYX\) that way, which is the required angle.
Can you take it from here?