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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.
Picture: http://www.math.sunysb.edu/~scott/mat360.spr04/cindy/incenter.gif
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Are we allowed to use calculus?
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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
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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
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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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