Math Is Fun Forum

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Gergonne triangle (fancy name from da wiki :D)

Let the incircle of triangle ABC be tangent to sides 

and

at D, E, and F, respectively. Prove that triangle DEF is acute.
I just have no idea how to do this. I think I should find the angles in terms of the larger triangles angles, but I duno how...:/
this thing is haaard. thanks!!!:D

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Good. You can read.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Gergonne triangle (fancy name from da wiki :D)

hi thedarktiger,

Like the avatar by the way.

Diagram below.

Some preliminaries.

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.

Bob

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Re: Gergonne triangle (fancy name from da wiki :D)

thanks!

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Good. You can read.