geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the Poncelet point of four given points is defined as follows: Let be four points in the plane that do not form an

orthocentric system In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through disjoint pairs among the points are perpendicular, and ...

and such that no three of them are

collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...

. The

nine-point circle 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 midpoint of eac ...

s of triangles meet at one point, the Poncelet point of the points . (If do form an orthocentric system, then triangles all share the same nine-point circle, and the Poncelet point is undefined.)

Properties

If do not lie on a circle, the Poncelet point of lies on the

circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every pol ...

of the

pedal triangle In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle. More specifically, consider a triangle ''ABC'', and a point ''P'' that is not one of the vertices ''A, B, C''. Drop perpendiculars from ''P'' to the th ...

of with respect to triangle and lies on the other analogous circles. (If they do lie on a circle, then those pedal triangles will be lines; namely, the

Simson line In geometry, given a triangle and a point on its circumcircle, the three closest points to on lines , , and are collinear. The line through these points is the Simson line of , named for Robert Simson. The concept was first published, howeve ...

of with respect to triangle , and the other analogous Simson lines. In that case, those lines still concur at the Poncelet point, which will also be the anticenter of the cyclic quadrilateral whose vertices are .) The Poncelet point of lies on the circle through the intersection of lines and , the intersection of lines and , and the intersection of lines and (assuming all these intersections exist). The Poncelet point of is the center of the unique rectangular hyperbola through .

References

* * Euclidean plane geometry {{elementary-geometry-stub