![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
is any point on the side
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
of the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
.
![r,r_1,r_2](/media/m/5/d/6/5d6a9a393b3152540bf8d04bb8291ed3.png)
are the radii of the circles inscribed in
![ABC,AMC,BMC](/media/m/c/a/7/ca72fe8d3704900a0d06ac8c170449f7.png)
.
![q](/media/m/c/1/d/c1db9b1124cc69b01f9a33595637de69.png)
is the radius of the circle on the opposite side of
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
to
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
, touching the three sides of
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and the extensions of
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
and
![CB](/media/m/d/d/7/dd7ebd02df3c940a7e47b8a09480e1b1.png)
. Similarly,
![q_1](/media/m/6/7/c/67c86efc63a51050f6ef0e84a91cec5f.png)
and
![q_2](/media/m/6/e/6/6e6f9ae8f824a18aa7d1970e0029987b.png)
. Prove that
![r_1r_2q=rq_1q_2](/media/m/6/2/0/6208ed6752951bedfd403d899ff968ec.png)
.
%V0
$M$ is any point on the side $AB$ of the triangle $ABC$. $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$. $q$ is the radius of the circle on the opposite side of $AB$ to $C$, touching the three sides of $AB$ and the extensions of $CA$ and $CB$. Similarly, $q_1$ and $q_2$. Prove that $r_1r_2q=rq_1q_2$.