Vladimir Nikolin

There are several proofs of the concurrency of the altitudes, this is yet another one.

As we know, if point F is antipode of the vertex C, so that CF is a diameter of the circumcircle of the triangle ABC and H is intersection of the two altitudes ha and hb to the sides a and b,

Attach:Geometry/altitudes.png Δ| CH is parallel to OMc

then (corollary 2), the triangle CFH has midline OMc, CH is parallel to OMc.

Line OMc is the perpendicular bisector of the side AB, which means that OMc \perp AB , so also CH \perp AB. Q.E.D.