Nine-point circle

Although he is credited for its discovery, Karl Wilhelm Feuerbach did not entirely discover the nine-point circle, but rather the six-point circle, recognizing the significance of the midpoints of the three sides of the triangle and the feet of the altitudes of that triangle.

(At a slightly earlier date, Charles Brianchon and Jean-Victor Poncelet had stated and proven the same theorem.)

But soon after Feuerbach, mathematician Olry Terquem himself proved the existence of the circle.

There are six "sidelines" in the quadrilateral; the nine-point conic intersects the midpoints of these and also includes the diagonal points.

The conic is an ellipse when P is interior to △ABC or in a region sharing vertical angles with the triangle, but a nine-point hyperbola occurs when P is in one of the three adjacent regions, and the hyperbola is rectangular when P lies on the circumcircle of △ABC.