Circle Gamma is the incircle of triangle ABC and is also the circumcircle of triangle XYZ. The point X is on segment BC, the point Y is on segment AB, and the point Z is on segment AC. If angle A=40 degrees,angle B=60 degrees, and angle C=80 degrees, what is the measure of angle YZX?

Guest Sep 12, 2018

#1**+1 **

See the following image :

AZDY forms a kite with angles AZD, AYD = 90

And ZAY = 40

So...angle ZDY = 360 - 2(90) - 40 = 140

And in triangle ZDY, because DZ = DY....then triangle ZDY is isosceles with angle YZD = [ 180 - 140 ] / 2 =

40/2 = 20

And CXDZ also forms a kite with angles DXC, DZC = 90

And angle ZCX = 80

So....angle ZDX = 360 - 2(90) - 80 = 100

And in triangle ZDX, because DX = DZ....then triangle ZDX is isosceles with angle DZX = [ 180 - 100 ] / 2 =

80/2 = 40

And angle YZX = angle YZD + angle DZX = 20 + 40 = 60

CPhill Sep 13, 2018