Given a point and a triangle , the Miquel triangle is the triangle connecting the side points , , and of with respect to which is the Miquel point.
Let the points defining the Miquel circles be fractional distances , , and along the sides , , and , respectively, and let and . The Miquel triangle has side lengths
(1)
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(2)
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(3)
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and area
(4)
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where is the area of the reference triangle.
In the special case , the Miquel triangle becomes the medial triangle.
All Miquel triangles of a given point are directly similar, and is the similitude center in every case.