AUSTRIAN-POLISH 2006 Problem 3

$ABCD$ is a tetrahedron. Let $K$ be the center of the incircle of $CBD$ . Let $M$ be the center of the incircle of $ABD$ . Let $L$ be the gravycenter of $DAC$ . Let $N$ be the gravycenter of $BAC$ .

Suppose $AK$ , $BL$ , $CM$ , $DN$ have one common point. Is $ABCD$ necessarily regular?