Let the incircle of triangle ABC be tangent to sides line BC, line AC, and line AB at D, E, and F, respectively. Prove that triangle DEF is acute.
Here's one way to do it, ACG..but maybe not the best...sorry I can't provide a pic...!!
Draw angle bisectors to A, B and C
Where these bisectors meet is the incenter of the triangle....and this will be the center of the circle....call this point, O
Now...looking at the circle, FE, FD and DE form chords within the circle....and any chord passing through O will be a diameter....but none of these chords will pass through O....and any chord in a circle not passing through the center will intercept an arc on the circle that is less than 180°
Thus....inscribed angles FDE, EFD and and FED will intercept arcs that will be less than 180°....so each of their measures will be less than (1/2) of 180° = less than 90°
So...triangle DEF is acute because each of its angles measure < 90°
I hope that helps ....!!!