In triangle ABC, BAC = 78 degrees and ACB = 33 degrees. The incircle of triangle ABC touches sides BC, AC, and AB at D, E, and F, respectively. Find EFD in degrees.
+128475 B
69
O D
F
A E C
78 33
Call the center of the incircle O
Angle BAC = 78°
Angles OFA and OEA = 90°
And AFOE is a quadrilateral
So angle FOE = 180 - 78 = 102°
Connect EF....
OF and OE are radii so triangle FOE is isoceles
So angle OFE = (180 - 102 ) / 2 = 78 / 2 = 39°
Angle ABC = 69°
And angles OFB and ODB = 90°
And FBDO is a quadrilateral
So angle FOD = 180 - 69 = 111°
Connect FD...
OF and OD are radii so triangle DFO is isoceles
So angle OFD = (180 - 111)/2 = 55.5°
So angle EFD = angle OFE + angle OFD = 39 + 55.5 = 94.5°
