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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 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?

 Jan 13, 2019
 #1
avatar+111437 
+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°

 

 

 

cool cool cool

 Jan 13, 2019
edited by CPhill  Jan 13, 2019
edited by CPhill  Jan 14, 2019
 #2
avatar+110116 
+2

Nice pic Chris :)

Melody  Jan 13, 2019
 #3
avatar+111437 
+1

Thanks, Melody...!!!!

 

.....it took me long enough to make it in GeoGebra......this utility is still kinda' funky to use...IMHO....

 

 

cool cool cool

CPhill  Jan 14, 2019
edited by CPhill  Jan 14, 2019
 #4
avatar+110116 
+1

Yes I have a lot of fun using Geogebra too. 

I am still finding new functions in there. It can really do a lot of things. 

 

For anyone that is interested,it is a free download.

Melody  Jan 14, 2019

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