Circle: finding locus of the chord's mid point Find the locus of the middle points of chords of the circle $$x^2+y^2 =a^2$$ which subtends a right angle at the point $(c, 0)$


Let $\Gamma$ be a circle centered at $O$ with radius $R$ and $P$ a point inside $\Gamma$. Let $ABCD$ a quadrilateral inscribed in $\Gamma$ with orthogonal diagonals $AC,BD$ meeting at $P$. By Thales' theorem the midpoints of $AB,BC,CD,DA$ are the vertices of a rectangle with sides parallel to $AC,BD$. Since $O$ is the circumcenter of $ABCD$, the lines joining $O$ with the previous midpoints are orthogonal to the sides of $ABCD$. By angle chasing those midpoints lie on a circle centered at the midpoint of $OP$, whose diameter equals $\frac{1}{2}\sqrt{BD^2+AC^2}$. Let $u,v$ be the distances of $O$ from $AC,BD$. By the Pythagorean theorem we have $u^2+v^2=OP^2$ and $\left(\frac{1}{2}AC\right)^2+u^2 = OA^2 = R^2$, hence 
$$\tfrac{1}{2}\sqrt{AC^2+BD^2}=\sqrt{2R^2-OP^2}$$
is constant and the midpoints of $AB,BC,CD,DA$ lie on a fixed circle centered at the midpoint of $OP$. This essentially is the Proposition $11$ from Archimedes' Book of Lemmas, which is used to prove the pizza theorem, too (if you are confident with Italian, you may have a look at page 21 of these notes, too).
