Question 116
Let $ABCD$ be a cyclic quadrilateral whose diagonals are not perpendicular and intersect at $X$. Let $A', C'$ be the projections of $A$ and $C$ onto the line $BD$ and let $B', D'$ be the projections of $B$ and $D$ onto $AC$. Prove that:
a) the perpendicular lines drawn from the midpoints of the sides onto the opposite sides are concurrent at a point called *Mathot's point*;
b) points $A', B', C', D'$ are cocyclic;
c) if $O'$ is the circumcenter of $A'B'C'$, then $O'$ is the midpoint of the line segment determined by the orthocenters of triangles $XAB$ and $XCD$;
d) $O'$ is the *Mathot point* of the quadrilateral $ABCD$.
a) the perpendicular lines drawn from the midpoints of the sides onto the opposite sides are concurrent at a point called *Mathot's point*;
b) points $A', B', C', D'$ are cocyclic;
c) if $O'$ is the circumcenter of $A'B'C'$, then $O'$ is the midpoint of the line segment determined by the orthocenters of triangles $XAB$ and $XCD$;
d) $O'$ is the *Mathot point* of the quadrilateral $ABCD$.