Poncelet point

{{short description|Intersection of the 9-point circles of all triangles made from 4 points}}

In geometry, the Poncelet point of four given points is defined as follows:

Let {{mvar|A, B, C, D}} be four points in the plane that do not form an orthocentric system such that no three of them are collinear. The nine-point circles of triangles {{math|△ABC, △BCD, △CDA, △DAB}} meet at one point, the Poncelet point of the points {{mvar|A, B, C, D}}. (If {{mvar|A, B, C, D}} do form an orthocentric system, then triangles {{math|△ABC, △BCD, △CDA, △DAB}} all share the same nine-point circle, and the Poncelet point is undefined.)

Properties

If {{mvar|A, B, C, D}} do not lie on a circle, the Poncelet point of {{mvar|A, B, C, D}} lies on the circumcircle of the pedal triangle of {{mvar|D}} with respect to triangle {{math|△ABC}} 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 of {{mvar|D}} with respect to triangle {{math|△ABC}}, 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 {{mvar|A, B, C, D}}.)

The Poncelet point of {{mvar|A, B, C, D}} lies on the circle through the intersection of lines {{mvar|AB}} and {{mvar|CD}}, the intersection of lines {{mvar|AC}} and {{mvar|BD}}, and the intersection of lines {{mvar|AD}} and {{mvar|BC}} (assuming all these intersections exist).

The Poncelet point of {{mvar|A, B, C, D}} is the center of the unique rectangular hyperbola through {{mvar|A, B, C, D}}.

References

{{citation|title=The Feuerbach point and reflections of the Euler line|journal=Forum Geometricorum|first=Jan|last=Vonk|url=http://forumgeom.fau.edu/FG2009volume9/FG2009Volume9.pdf#page=51|volume=9|year=2009|pages=47–55}}
{{citation|title=Poncelet points and antigonal conjugates|url=https://artofproblemsolving.com/community/c6h109112}}

Category:Euclidean plane geometry