Editing Nine-point circle (section)

==Generalization==
{{Main|Nine-point conic}}
The circle is an instance of a [[conic section]] and the nine-point circle is an instance of the general nine-point conic that has been constructed with relation to a triangle {{math|△''ABC''}} and a fourth point {{mvar|P}}, where the particular nine-point circle instance arises when {{mvar|P}} is the orthocenter of {{math|△''ABC''}}. The vertices of the triangle and {{mvar|P}} determine a [[complete quadrilateral]] and three "diagonal points" where opposite sides of the quadrilateral intersect. 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 {{mvar|P}} is interior to {{math|△''ABC''}} or in a region sharing [[vertical angles]] with the triangle, but a [[nine-point hyperbola]] occurs when {{mvar|P}} is in one of the three adjacent regions, and the hyperbola is rectangular when P lies on the circumcircle of {{math|△''ABC''}}.