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.
vladimir.shelomovskii@gmail.com, vvsss
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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