Question 75
Consider a triangle $ABC$ and a point $M$ in its interior. Denote by $D, E, F$ the projections of $M$ onto the sides $BC, CA$ respectively $AB$ and by $r$ the radius of its incircle. Prove that if
$$
\frac{BC}{MD} + \frac{CA}{ME} + \frac{AB}{MF} = \frac{AB + BC + CA}{r},
$$
then the straight lines $AD$, $BE$ and $CF$ are concurrent.
$$
\frac{BC}{MD} + \frac{CA}{ME} + \frac{AB}{MF} = \frac{AB + BC + CA}{r},
$$
then the straight lines $AD$, $BE$ and $CF$ are concurrent.