Math forum

TomasT
 #1 Posted : Friday, May 06, 2011 6:41:07 PM
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A triangle has two sides of length 1. Determine the length of the third side in order to maximize the radius, R, of a circle that is inscribed in the triangle.

Andy
 #2 Posted : Wednesday, October 12, 2011 1:04:32 AM
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Are we allowed to use calculus?

12UK5
 #3 Posted : Tuesday, November 08, 2011 4:22:11 PM
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nerokid
 #4 Posted : Wednesday, November 16, 2011 3:50:41 PM
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call O the center of the circle inscribed in the triangle , H is the  centrol point of AB , because CA = CB we have C,O,H is on the same line , because the bisector of AO is the bisector of BAC so we have OH /OC = AO/AC and OH + OC = CH , CH = AB/cos(BCO) , and AO = AB/2 so we can have the result OH and OC in term of AB and BCO , note that R = OH * sinOCB , by replacing OH and note that we can calculate sin OCB and cos OCB in term of AB so we get R = F(AB) , all u need to is find the maximum off this function of AB >0

vinay
 #5 Posted : Tuesday, December 20, 2011 3:42:40 AM
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Simply put, the triangle needs to be equilateral to get the maximum radius of the circle inside it.

Hence the third side also needs to be 1

Gyanesh Kumar Verma
 #6 Posted : Friday, May 11, 2012 10:17:40 AM
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Suppose midpoint of AB is H

Angle ABI(I stands for incenter )=ф

Angle BCH =Ө

As it is clear 2ф+Ө=π/2………………………………….(0)

BH=CB sinӨ

r = BH tanф

r = CB sinӨ tanф

r =1*sin(π/2-2ф)tanф

r=cos2фtanф…………………….(1)

using a little calculas (maxima and minima) for maximum r

ф=π/8 and using equation (0)   Ө=π/4

and radius r = (1-sin π/4)

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