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
(2|AM|)^2 = 2|AB|^2
4|AM|^2 = 2(|AL| + |LB|)^2
= 2(|AM| + r)^2
or
0 = 2|AM|^2 - 4r|AM| - 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 2007-08-20 11:23:46 |
Solution
