I (radius=r) is the incircle of triangle ABC (right angled at B) and O (radius=R) is another circle which touches the extended side BC and AC. Find the ratio (R/r) if the point of intersection of circles I and O is midpoint of AC.
If the incircle of triangle ABC touches the
midpoint of the hypotenuse, then ABC is a
isoceles right triangle.
Let L and M be the points where the incircle
touches sides AB and AC rspectively.
AC^2 = AB^2 + BC^2
(2AM)^2 = 2AB^2
4AM^2 = 2(AL + LB)^2
= 2(AM + r)^2
or
0 = 2AM^2  4rAM  2r^2
or
AM = r(1 + sqrt(2))
From similar triangles OMA and AMC we get
OM AM
 = 
AM IM
or
rR = AM^2
or
R
 = 3 + 2sqrt(2) ~= 5.828427
r

Posted by Bractals
on 20070820 11:23:46 