MathDB

The tangents to the inscribed circle of

\triangle ABC

, which are parallel to the sides of the triangle and do not coincide with them, intersect the sides of the triangle in the points

M,N,P,Q,R,S

such that

M,S\in (AB)

N,P\in (AC)

Q,R\in (BC)

. The interior angle bisectors of

\triangle AMN

\triangle BSR

and

\triangle CPQ

, from points

A,B

and respectively

C

have lengths

l_{1}

l_{2}

and

l_{3}

.\\ Prove the inequality:

\frac {1}{l^{2}_{1}}+\frac {1}{l^{2}_{2}}+\frac {1}{l^{2}_{3}} \ge \frac{81}{p^{2}}

where

p

is the semiperimeter of

\triangle ABC

Problem Statement