Pivot theorem
Encyclopedia
In geometry
, the pivot theorem states that, given any three points P, Q, and R on each respective side of a triangle ABC, the three circles through the points AQR, BPR and CPQ share a common point M. Conversely, this is equivalent to a porism
: given any three circles sharing a common point M, there are an infinite number of triangles such that one point lies on each circle and the sides of the triangle pass through the intersection points of the circles.
There is also a three-dimensional analog, in which the four spheres passing through a point of a tetrahedron and points on the edges of the tetrahedron intersect in a common point.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the pivot theorem states that, given any three points P, Q, and R on each respective side of a triangle ABC, the three circles through the points AQR, BPR and CPQ share a common point M. Conversely, this is equivalent to a porism
Porism
A porism is a mathematical proposition or corollary. In particular, the term porism has been used to refer to a direct result of a proof, analogous to how a corollary refers to a direct result of a theorem.-Beginnings:...
: given any three circles sharing a common point M, there are an infinite number of triangles such that one point lies on each circle and the sides of the triangle pass through the intersection points of the circles.
There is also a three-dimensional analog, in which the four spheres passing through a point of a tetrahedron and points on the edges of the tetrahedron intersect in a common point.