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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
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 #1
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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.       sad

I shall report it as a problem :/

Melody  Sep 27, 2017
edited by Melody  Sep 27, 2017
 #2
avatar+79784 
+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°

 

 

 

 

 

cool cool cool

CPhill  Sep 28, 2017
edited by CPhill  Sep 28, 2017
edited by CPhill  Sep 28, 2017
edited by CPhill  Sep 28, 2017

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