proof of pivot theorem
Let be a triangle, and let , , and be points
on , , and , respectively. The circumcircles of
and intersect in and in another
point, which we call . Then and are cyclic
quadrilaterals
, 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, the circumcircles of the triangles , , and have a common point, .
Title | proof of pivot theorem |
---|---|
Canonical name | ProofOfPivotTheorem |
Date of creation | 2013-03-22 13:12:37 |
Last modified on | 2013-03-22 13:12:37 |
Owner | pbruin (1001) |
Last modified by | pbruin (1001) |
Numerical id | 4 |
Author | pbruin (1001) |
Entry type | Proof |
Classification | msc 51M04 |