[[File:EulerCircle4.gif|thumb| Note that the construction still works even if the orthocenter and circumcenter fall outside of the triangle.]] In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: The nine-point circle is also known as Feuerbach's circle (after Karl Wilhelm Feuerbach), Euler's circle (after Leonhard Euler), Terquem's circle (after Olry Terquem), the six-points circle, the twelve-points circle, the n-point circle, the medioscribed circle, the mid circle or the circum-midcircle. Its center is the nine-point center of the triangle.

Nine significant points

The diagram above shows the nine significant points of the nine-point circle. Points D, E, F are the midpoints of the three sides of the triangle. Points G, H, I are the feet of the altitudes of the triangle. Points J, K, L are the midpoints of the line segments between each altitude's vertex intersection (points A, B, C) and the triangle's orthocenter (point S). For an acute triangle, six of the points (the midpoints and altitude feet) lie on the triangle itself; for an obtuse triangle two of the altitudes have feet outside the triangle, but these feet still belong to the nine-point circle.

Discovery

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. (See Fig. 1, points D, E, F, G, H, I.) (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. He was the first to recognize the added significance of the three midpoints between the triangle's vertices and the orthocenter. (See Fig. 1, points J, K, L.) Thus, Terquem was the first to use the name nine-point circle.

Tangent circles

In 1822 Karl Feuerbach discovered that any triangle's nine-point circle is externally tangent to that triangle's three excircles and internally tangent to its incircle; this result is known as Feuerbach's theorem. He proved that:"... the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle..." The triangle center at which the incircle and the nine-point circle touch is called the Feuerbach point.

Other properties of the nine-point circle

Figure 3 Figure 4 ω be the nine-point circle of the diagonal triangle of a cyclic quadrilateral. The point of intersection of the bimedians of the cyclic quadrilateral belongs to the nine-point circle. [[File:Nine-point circle of diagonal triangle.png|thumb|ABCD is a cyclic quadrilateral. △EFG is the diagonal triangle of ABCD. The point T of intersection of the bimedians of ABCD belongs to the nine-point circle of △EFG .]] A, B, C, D are given that do not form an orthocentric system, then the nine-point circles of △ABC, △BCD, △CDA, △DAB concur at a point, the Poncelet point of A, B, C, D. The remaining six intersection points of these nine-point circles each concur with the midpoints of the four triangles. Remarkably, there exists a unique nine-point conic, centered at the centroid of these four arbitrary points, that passes through all seven points of intersection of these nine-point circles. Furthermore, because of the Feuerbach conic theorem mentioned above, there exists a unique rectangular circumconic, centered at the common intersection point of the four nine-point circles, that passes through the four original arbitrary points as well as the orthocenters of the four triangles. A, B, C, D are given that form a cyclic quadrilateral, then the nine-point circles of △ABC, △BCD, △CDA, △DAB concur at the anticenter of the cyclic quadrilateral. The nine-point circles are all congruent with a radius of half that of the cyclic quadrilateral's circumcircle. The nine-point circles form a set of four Johnson circles. Consequently, the four nine-point centers are cyclic and lie on a circle congruent to the four nine-point circles that is centered at the anticenter of the cyclic quadrilateral. Furthermore, the cyclic quadrilateral formed from the four nine-pont centers is homothetic to the reference cyclic quadrilateral ABCD by a factor of –1⁄2 and its homothetic center N lies on the line connecting the circumcenter O to the anticenter M where x : y : z be a variable point in trilinear coordinates, an equation for the nine-point circle is

Generalization

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 △ABC and a fourth point P, where the particular nine-point circle instance arises when P is the orthocenter of △ABC . The vertices of the triangle and 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 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 .