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**+2 **

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°

CPhill Jul 25, 2017