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 $\angle AYX$?

michaelcai
Sep 27, 2017

#1**0 **

Circle \;\;\Gamma\;\; \text{ is the incircle of }\triangle ABC \text{ and is also the circumcircle of }\triangle XYZ\text{ The point X is on }\overline{BC} \text{ point Y is on }\overline{AB}, \textand the point Z is on }\overline{AC}.\;\; If\;\; $\angle A=40^\circ, \;\;\angle B=60^\circ,\;\; and \;\;\angle C=80^\circ, \text{what is the measure of }\angle AYX\; ?

\(\text{Circle }\Gamma\;\; \text{ is the incircle of } \triangle ABC \text{ and is also the circumcircle of } \triangle XYZ\\ \text{ The point X is on }\overline{BC} \text{and the point Y is on }\overline{AB}, \\ \text{and the point Z is on }\overline{AC}.\\ If\;\; \angle A=40^\circ, \;\;\angle B=60^\circ,\;\; and \;\;\angle C=80^\circ, \text{what is the measure of }\angle AYX\; ? \)

Let O be the centre of the circle.

OX=OY equal radio

So

OXY is an isosceles triangle

120+2

so

< AYX =

I am sorry, this was a full answer but 3/4 of it has been deleted.

There is obviously a software problem.

I shall report it as a problem :/

Melody
Sep 27, 2017

#2**+1 **

See the following image :

Construct angle bisectors of each vertex angle of triangle ABC

Angle AYC = 180 - angle ACY - angle YAC = 180 - 40 - 40 = 100°

Angle AOC = 180 - angle OAC - angle OCA = 180 - 20 - 40 = 120°

And angle YOX is a vertical angle to angle AOC ....so it measures 120°

And since they are equal radii, OY = OX

So angle OYX = angle OXY

So triangle OYX is isosceles

And.....angle OYX = [ 180 - 120 ] / 2 = 60 / 2 = 30°

And AYX = angle AYC + angle OYX = 100 + 30 = 130°

CPhill
Sep 28, 2017