Show that two segments are perpendicular

Question

Let $ABC$ be a triangle such that $|AC|=|BC|.$ Let $M$ be the midpoint of $AB$ and let $D$ be the midpoint of $MC$. Let $S$ be the the point obtained from projecting $M$ orthogonally onto $AD.$ Show that $BS$ and $CS$ are perpendicular.

My idea: Show that you can inscribe a quadrilateral $CSMB$ in a circle. That way the angle $CSB$ equals the angle $CMB.$ To show that you you can inscribe $CSMB$ in a circle, it is enough to show one of the following:

1. the angle $SCM$ equals the angle $SBM$
2. The angle $MSB$ equals the angle $MCB$
3. The angle $MCB$ equals the angle $MSB$

Any help would be greatly appreciated.

Answer 1

It's easy to see that angle $SMB = SDC$, and $SM/MA = SD/DM$, that is $SM/MB = SD/DC$. Hence triangles $SMB, SDC$ are similar. Hence angles $BSM, CSD$ are equal. Hence angle $CSB$ equals angle $DSM$, which is a right angle by construction.