Ratios Involving Six Radii

move red point M
labels

Given a point

M

on the side

AB

of the triangle

ABC

, let

r

1

and

r

2

be the radii of the inscribed circles of the triangles

ACM

and

BCM

, and let

ρ

1

and

ρ

2

be the radii of the circumcircles of the triangles

ACM

and

BCM

. Let

r

and

ρ

denote the radii of the inscribed circle and the circumcircle of the triangle

ABC

, respectively. Prove that

r

1

r

2

ρ

1

ρ

2

=

r

ρ

.