Nine Point Circle - Proof #3

This proof requires a Lemma. Let X be a point on a given circle of center O and radius r. Let H be any given fixed point. Then the locus of the point Y which is the midpoint of HX is a circle with radius r/2 and center N, where N is the midpoint of HO.

For by similar triangles, the length of YN is r/2.

Now the applet will lead you through the third proof.