What are the measures of the apex angles of all isosceles triangles with circumradius 9 and inradius 4?

(As a reminder, the circumradius is the radius of the circumscribed circle, and the inradius is the radius of the inscribed circle.)

The MathWorld article on "Inradius" gives the neat formula

r=R(cosA+cosB+cosC-1)

for any triangle where r is the inradius and R is the circumradius.  Specializing to an isosceles triangle with apex angle C gives

r=R(2sinC/2+cosC-1)=R(2sqrt(1-cosC)/2+cosC-1).

Setting r=4 and R=9, this then leads to the quadratic in x=1-cosC

x^2-(10/9)x+16/81=0

which has the roots 8/9 and 2/9 so that cosC=1/9 or 7/9.

Posted by Richard on 2006-07-05 22:00:40