Let the incircle of triangle ABC be tangent to sidesand at D, E, and F, respectively. Prove that triangle DEF is acute.
Good. You can read.
Like the avatar by the way.
The centre O is on the intersection of the angle bisectors.
In triangles AFO and AEO, AO is common, OF = OE = radius and FAO = EAO so these are congruent. Therefore, AF = AE.
Similarly, BF = BD and CE = CD.
So triangles AEF, BFD and CDE are each isosceles.
Now to get an expression for angle EFD.
AFE = 90 - A/2 and BFD = 90 - B/2
=> EFD = 180 - (90 - A/2 + 90 - B/2) = (A + B)/2 = (180 - C)/2 = 90 - C/2
As C < 180 => C/2 < 90 => EFD is acute.
A similar argument can be used for the other two angles of DEF.
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei