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Let the incircle of triangle ABC be tangent to sides BC, AC, and AB at D, E, and F, respectively. Prove that triangle DEF is acute.

 

Guest Mar 13, 2017
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 #1
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Here's one method:

 

Find the center of the circle and call it "O"

 

Draw diameter F' D' through O  such that F'D' is parallel to FD

 

Draw F'E  and D'E .....

 

Now triangle F'ED'  is a right triangle with angle F'ED'  being the right angle [ since it intercepts the diameter ]

 

But the diameter in any circle is the greatest chord possible....so FD is shorter than F'D'

 

Therefore.....angle FED intercepts a lesser arc than angle F'ED'....therefore, angle FED < angle F'ED'

 

And since angle F'ED'  was shown to be right, then angle FED must be < 90°....therefore, acute

 

And similarly,  by drawing diameter F'E'  parallel to FE and  E'D' parallel to ED, we can show that angles FDE and EFD  are also acute

 

 

cool cool cool

CPhill  Mar 13, 2017
 #2
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I drew the diagram but it is kind of onfusing on how you labeled the angles. Can you provide a diagram?

Guest Mar 13, 2017
 #3
avatar+75279 
+5

Here you go......it won't look exactly like the diagram, but it's close enough for you to get the idea

 

If you need extra explanation, let me know......

 

 

cool cool cool

CPhill  Mar 13, 2017

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