Three circles of different radius (C1,C2 and C3) touches each other as shown in the following diagram.
http://i233.photobucket.com/albums/ee201/vcpandya/…
The fourth Circle (red one) is then drawn. It is found that center of that fourth circle is Incenter of the triangle created by joining center of the other three (big) circles.
Find the ratio of C1:C2:C3
Bonus Question
What would be the answer if the center of fourth circle is
Case 1) Circumcenter
Case 2) Centroid
Case 3) Orthocenter
of the triangle created by joining center of the other three (big) circles.
1 Answer
? Favorite Answer
It seems to be impossible if radii are different.
If sides of triangle are a, b, c, then:
C1+C2 = a
C2+C3 = b
C3+C1 = c
or:
C1 = (a-b+c)/2
C2 = (a+b-c)/2
C3 = (-a+b+c)/2
These radii are the same as segments that we have on sides, as we draw incircle with radius r.
Then it must be:
(C1+C)^2 = C1^2 + r^2 => 2*C1*C+C^2 = r^2
But also 2*C2*C+C^2 = r^2
and 2*C3*C+C^2 = r^2
If we subtract any two equations (ie 1 and 2) it will be:
2*(C1-C2)*C = 0
It’s possible only if C1=C2, so ratio must be 1:1:1