# Locus of the midpoints of four chords

Author:
mweissa2
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.