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# Pls help, repost, last post guest gave an answer, no explanation

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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 $\angle AYX$?

Feb 9, 2021

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Angle AYX = 120 degrees

Feb 9, 2021
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Here is the diagram, and I have introduced a new point, E

BX and BY are both tangents to the circle, Hence they are the same length.

So  triangle EYB and triangle EXB are congruent.

So angle YEB = 90 degrees.

So angle EYB = 180-90-30 = 60 degrees

So angle AYX = the supplement of 60 degrees,  180-60 = 120 degrees

Feb 9, 2021
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Very nice, Melody   !!!!

CPhill  Feb 9, 2021
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Thanks Chris.

Melody  Feb 9, 2021
#6
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Thank you Melody! I got 120 later from old posts other people got, but not quite the same question. Your explanation is brilliant

Feb 9, 2021
#8
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You are welcome.

I should have seen the answer straight away but I didn't.

Maybe I will next time....

anyway, I am glad I could help.

Melody  Feb 9, 2021