The Pivot theorem is a special case of Miquel's theorem. It is trivial to prove but is a really surprising result. It states the following:
Let , , be the points on sides of , respectively. Then circles and pass through a common point called the Miquel point.
Let , , be the points on sides , , of (or even their extensions), respectively. The Miquel point lies on the circumcircle of if and only if points are collinear.
If you have difficulties in identifying the circles, take the circumcircle of a vertex and the two chosen points lying on the two sides through (adjacent to) that vertex.
The Pivot theorem is the case when , , respectively in the theorem above are not collinear. This result can be proved only by chasing a few angles as follows:
Consider the above configuration. Suppose that circles and intersect at . So quadrilaterals and are cyclic, meaning
giving Since also equals from the circle passing through points must also pass through point so that quadrilateral is also cyclic, i.e. circle passes through . So, is the Miquel point, and the proof is complete.
Consider the above configuration. To chase angles, we'll use directed angles to avoid configuration issues. Suppose that circles and intersect at . Then quadrilaterals and are cyclic, meaning and . Now just track that
So quadrilateral is also cyclic, that is, circle passes through . So is the Miquel point, and the proof is complete.