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?

Lightning Jan 13, 2019

#1**+2 **

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°

CPhill Jan 13, 2019