Let $M$, $N$ be midpoints of segments (respectively) $AD,BC$. $Q,P$ are orthogonal projections of points $N,M$ lines $AD,BC$. Assume circle with diameter $BC$ is tangent to line $AD$ in point $Q$. Then $NQ=NC$ as a radius. $MN||CD\implies$ area of triangles $MNC,MND$ is equal. Hence $MD\cdot CN=CN\cdot MP\implies MD=MP$. That proves circle with diameter $AD$ is tangent to line $BC$ in point $P$.[\hide]