In geometry, the nine point circle is a circle that can be constructed for any given triangle. It is named so because it passes through nine significant points, with six of them lying on the triangle itself: Geometry (Greek geo = earth, metro = measure) {put in greek letters here, check accuracy} arose as the field of knowledge dealing with spatial relationships. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

• the midpoints of the three sides,
• the feet of the altitudes,
• the midpoints of the portion of altitude between the vertices and the orthocenter.

It is also known as Feuerbach's circle, Euler's circle, Terquem's circle, six-points circle, twelve-points circle, n-point circle, medioscribed circle, mid circle or circum-midcircle. In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i. ... In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i. ...

Nine point circle (*.png version) File links The following pages link to this file: Nine point circle Categories: GFDL images ...

In the diagram above, the points are:

D, E, F - the midpoints of the three sides

G, H, I - the feet of the altitudes

J, K, L - the points on each altitude midway between the vertex and the orthocentre (labelled S)

The nine point circle is tangent externally to the three excircles and tangent internally to the incircle of the triangle, a theorem discovered by Karl Wilhelm Feuerbach in 1822 in the form: In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. ... In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. ... Karl Wilhelm Feuerbach. ... 1822 was a common year starting on Tuesday (see link for calendar). ...

... 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 following image illustrates this theorem:

9 point circle and excircles File links The following pages link to this file: Nine point circle Talk:Nine point circle Categories: GFDL images ...

The point at which the incircle and the nine point circle touch is often called the Feuerbach point.

Feuerbach was not the first to discover the circle. At a slightly earlier date, Charles Brianchon and Jean-Victor Poncelet had stated and proven the same theorem. Soon after Feuerbach, mathematician Olry Terquem also proved what Feuerbach did and added the three points that are the midpoints of the altitude between the vertices and the orthocenter. Terquem was the first to use the name nine point circle (as he was the first to associate nine special points with the circle). Charles Julien Brianchon (1783-1864) was a French mathematician and chemist. ... Jean-Victor Poncelet (July 1, 1788 – December 22, 1867) was a mathematician and engineer who did much to revive projective geometry. ... Olry Terquem (1782 - 1862) was a French mathematician, best known for his work in geometry, where he proved the Feuerbachs theorem about the nine point circle of a given triangle. ...

Other facts of interest:

• The radius of the nine point circle is half the length of the radius of the circumcircle of the triangle.

In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ... 9 point circle and circumcircle File links The following pages link to this file: Nine point circle Categories: GFDL images ...

• The nine point circle bisects any line from the orthocenter to a point on the circumcircle.

9 point circle bisection of lines from orthocenter of the triangle File links The following pages link to this file: Nine point circle Categories: GFDL images ...

• The center of the nine point circle (the nine point center) lies on the triangle's Euler line, at the midpoint between the triangle's orthocenter and circumcenter.
• If an orthocentric system of four points is given, any three of them define a triangle, and these four triangles all have the same nine point circle.
• The centers of the incircle and excircles of a triangle form an orthocentric system. The nine point circle created for that orthocentric system is the circumcircle of the original triangle.

In geometry, Eulers line (red line in the image) is the line passing through the orthocenter (blue), the circumcenter (green), the centroid (yellow), and the center of the nine point circle (red point) of any triangle. ... In geometry, an orthocentric system is a set of four points in the plane where one point is the orthocenter of the triangle formed by the other three. ...

See also

• synthetic geometry

Synthetic geometry is a descriptive term that identifies a methodology of geometry which makes use of theorems and synthetic observations to create theorems or solve problems, as opposed to analytic geometry which uses algebra, numbers, computations to draw theorems or solve problems. ...

External link

• Nine Point Centerby Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
• History about the nine point circle based on J.S. MacCay's article from 1892: History of the Nine Point Circle

• Nine Point Circle in Java
• Feuerbach's Theorem: a Proof (requires Java)