The incircles of $ABC$ and $ADC$ touch $AC$ at a point $P$. and the excircles of the same triangles touch the segment $AC$ at a point $Q$. Let $PK$ and $PL$ be diameters in the incircles of $ABC$ and $ADC$ respectively. Then in force of homothety the points $B, K, Q$ are collinear and the same is true about $D, L, Q$. Let the rays $BK$ and $DL$ intersect first time the incircle of $ABCD$ at $K_1$ and $L_1$ respectively. Again in force of homothety $K_1L_1$ is diameter in the incircle of $ABCD$. Also $K$ lies strictly inside the segment $K_1B$ thus $K_1L_1<KL=KP+PL$ which is equivalent to the statement of the problem.