Miquel's point
Contents
[hide]Miquel and Steiner's quadrilateral theorem
Let four lines made four triangles of a complete quadrilateral. In the diagram these are
Prove that the circumcircles of all four triangles meet at a single point.
Proof
Let circumcircle of circle cross the circumcircle of circle at point
Let cross second time in the point
is cyclic
is cyclic
is cyclic
is cyclic and circumcircle of contain the point
Similarly circumcircle of contain the point as desired.
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Circle of circumcenters
Let four lines made four triangles of a complete quadrilateral. In the diagram these are
Prove that the circumcenters of all four triangles and point are concyclic.
Proof
Let and be the circumcircles of and respectively.
In
In
is the common chord of and
Similarly, is the common chord of and
Similarly, is the common chord of and
points and are concyclic as desired.
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Triangle of circumcenters
Let four lines made four triangles of a complete quadrilateral.
In the diagram these are
Let points and be the circumcenters of and respectively.
Prove that and perspector of these triangles point is the second (different from ) point of intersection where is circumcircle of and is circumcircle of
Proof
Quadrungle is cyclic
Spiral similarity sentered at point with rotation angle and the coefficient of homothety mapping to , to , to
are triangles in double perspective at point
These triangles are in triple perspective are concurrent at the point
The rotation angle to is for sides and or angle between and which is is cyclic is cyclic.
Therefore is cyclic as desired.
Similarly, one can prove that
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Analogue of Miquel's point
Let inscribed quadrilateral and
points be given.
Prove that points and are concyclic.
Proof
Corollary
The points and are concyclic.
The points and are concyclic.
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Six circles crossing point
Let point point be given.
Denote tangent to tangent to
Prove that the circles and have the common point.
Proof
Let points and are concyclic, Similarly is the Miquel point of quadrungle is tangent to Similarly, is tangent to
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