proof of pivot theorem
Let be a triangle, and let , , and be pointson , , and , respectively. The circumcircles of and intersect in and in anotherpoint, which we call . Then and are cyclicquadrilaterals
, so
and
Combining this with and , we get
This implies that is a cyclic quadrilateral as well, so that lies on the circumcircle of . Therefore, thecircumcircles of the triangles , , and have a commonpoint, .