nine-point circle

The diagram above shows the nine significant points of the nine-point circle.

These four possible triangles will all have the same nine-point circle.

The nine-point circles form a set of four Johnson circles.

The triangle's nine-point circle has half the diameter of the circumcircle.

The orthopole of lines passing through the circumcenter lie on the nine-point circle.

The nine-point circle created for that orthocentric system is the circumcircle of the original triangle.

In geometry, the nine-point circle is a circle that can be constructed for any given triangle.

Karl Feuerbach describes the nine-point circle of a triangle.

The point where the nine-point circle touches the incircle is known as the Feuerbach point.

The circumcircle is tangent to the nine-point circle.

The center of this common nine-point circle lies at the centroid of the four orthocentric points.

The point at which the incircle and the nine-point circle touch is often referred to as the Feuerbach point.

Several other sets of points defined from a triangle are also concyclic, with different circles; see nine-point circle and Lester's theorem.

A nine-point circle bisects a line segment going from the corresponding triangle's orthocenter to any point on its circumcircle.

The center of all rectangular hyperbolas that pass through the vertices of a triangle lies on its nine-point circle.

The remaining six intersection points of these nine-point circles each concur with the midpoints of the four triangles.

The nine-point circle also passes through the three orthogonal intersections at the feet of the altitudes of the four possible triangle.

(The same 'midpoints' defining separate nine-point circles, those circles must be concurrent.)

The nine-point circles are all congruent with a radius of half that of the cyclic quadrilateral's circumcircle.

The orthocenter, along with the centroid, circumcenter and center of the nine-point circle all lie on a single line, known as the Euler line.

It passes through the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.

The nine-point circle of the general triangle has an analogue in the circumsphere of a tetrahedron's medial tetrahedron.

Feuerbach's theorem states that the nine-point circle is tangent to the incircle and the three excircles of a reference triangle.

The radius of a triangle's circumcircle is twice the radius of that triangle's nine-point circle.

The circle tangent to all three of the excircles as well as the incircle is known as the nine-point circle.