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#1 2014-02-01 21:53:48

thedarktiger
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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...hmm
this thing is haaard. thanks!!!big_smile


Batman shows Superman his new phone. Batman-"Alfred, whats the temperature in London?"
Alfred-"Just a second, sir." Alfred turns on his iPhone."Siri, whats the temperature in London?"
Siri - "Six centigrade." Alfred-"Six centigrade sir." Batman-"See?"

#2 2014-02-02 01:33:37

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

hi thedarktiger,

Like the avatar by the way. up

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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#3 2014-02-03 20:49:01

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

thanks!


Batman shows Superman his new phone. Batman-"Alfred, whats the temperature in London?"
Alfred-"Just a second, sir." Alfred turns on his iPhone."Siri, whats the temperature in London?"
Siri - "Six centigrade." Alfred-"Six centigrade sir." Batman-"See?"
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