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In triangle $ABC$, \(\angle BAC = 72^\circ\). The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively. Find $\angle EDF$, in degrees.

 

Link to diagram:

 

https://latex.artofproblemsolving.com/0/0/a/00af01b8929e78fd074011c2a75baf7d2f45f0a6.png

 

can anyone tell me how to put a diagram onto web2.0calc.com?

michaelcai  Jul 25, 2017
 #1
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Note that if we called the center of the circle, O, then radii drawn to both tangent points F and and E would form right angles AEO and  AFO......so.....FAEO  would form a quadrilateral such that  angle FOE would be suppllemental to angle  FAE  = angle BAC .

 

So....angle FOE  = 180 - 72 =  108°

 

And since FOE is a central angle intercepting the same arc as the inscribed angle FDE, then  FDE  = (1/2) FOE  = (1/2) 108  =  54°

 

 

cool cool cool

CPhill  Jul 25, 2017
edited by CPhill  Jul 25, 2017

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