Circle Gamma is the incircle of triangle ABC and is also the circumcircle of triangle XYZ. The point X is on BC, the point Y is on AB, and the point Z is on AC. If angle A=40 degrees, angle B=60 degrees, and angle C=80 degrees, what is the measure of angle YZX?
See the following image :
Let D be the center of the circle
Since BA and BC are tangents meeting radii DX and DY.....then....
Angle BYD = Angle BXD = 90°
And angle YBX = 60°
And XBYD is a quadrilateral.....so the sum of its interior angles = 360
So angle YDX = 360 - 90 - 90 - 60 = 120°
And YDX is a central angle in the circle
And angle YZX is an inscribed angle in the circle intercepting the same arc as angle YDX
So....its measure is (1/2)YDX = 60°