This dynamic geometry sketch is intended to accompany the question posed at http://math.stackexchange.com/questions/1610903/find-the-locus-of-the-midpoint. In the diagram we see a circle (centered at $O$) and an interior point $P$. Two perpendicular lines (shown as dashed in the diagram) intersect at $P$, and intercept the circle in four points, which are joined by four chords (shown as boldfaced segments). So each of the four boldfaced chords subtends a $90°$ angle at $P$. The midpoints of the four chords are $W,X,Y$ and $Z$, shown in red.
The orientation of the two perpendicular lines can be changed by dragging point $Q$ (purple) around the circumference of the circle. As $Q$ moves, the chords and their midpoints move as well. So the question is:
As $Q$ varies, how do the red points move? What path is traced out by $WX,Y$ and $Z$?
The “Show/Hide Locus” toggle button reveals the path. It appears to be a circle! More precisely, it appears to be a circle centered at the midpoint $M$ of the segment joining $O$ to $P$. You can reveal point $M$ (shown in green) with the second toggle button.
I don’t, unfortunately, have a proof for you of why this is so, but now at least you know what you need to prove.