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Incenter and a half (Posted on 2018-02-03) Difficulty: 3 of 5
Take an isosceles triangle and its inradius r1. Split the triangle by its altitude. The radius of one of these smaller triangles is r2.

If r1/r2=3/2 what is the ratio of base to height in the original triangle?

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The inradius is a perpendicular to each side of a triangle so Area = 1/2 * inradius * (sum of sides).  

If the isosceles triangle has sides (a,a,2b) then its area = 1/2 * r1 * (2a+2b) = twice area of split triangle = 2 * 1/2 * r2 * (a+b+h). Given r1/r2 =3/2, this reduces to a+b=2h.

From the half-triangle b^2 + h^2 = a^2.  Substitute a=b-2h to get 4b=3h and 2b/h = 3/2.

  Posted by xdog on 2018-02-03 16:06:13
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