Let
be a convex quadrilateral with
different from
. Denote the incircles of triangles
and
by
and
respectively. Suppose that there exists a circle
tangent to ray
beyond
and to the ray
beyond
, which is also tangent to the lines
and
.
Prove that the common external tangents to
and
intersects on
.
Author: Vladimir Shmarov, Russia
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
![BA](/media/m/f/3/e/f3ee5efe9b25bd27cd3ada1235d36017.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![ADC](/media/m/b/5/b/b5ba4f2dbf8650a6779ccd5923a7f007.png)
![k_{1}](/media/m/e/0/0/e00d670f2a0f9034eb624c89c6cbbaae.png)
![k_{2}](/media/m/d/4/2/d42853c656fc137c156d33c994da39ca.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![BA](/media/m/f/3/e/f3ee5efe9b25bd27cd3ada1235d36017.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
Prove that the common external tangents to
![k_{1}](/media/m/e/0/0/e00d670f2a0f9034eb624c89c6cbbaae.png)
![k_{2}](/media/m/d/4/2/d42853c656fc137c156d33c994da39ca.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
Author: Vladimir Shmarov, Russia