Trigonometry/Circles and Triangles/The Miquel Point

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In general, four straight lines define four triangles (by omitting each line in turn we have three lines forming a triangle).

The four circumcircles of these triangles all intersect at the Miquel point. This point and the four circumcentres all lie on a circle.

The unique parabola of which the four lines are all tangents has its focus at the Miquel point.