Miquel's point
Miquel and Steiner's quadrilateral theorem
500px|right
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
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.
vladimir.shelomovskii@gmail.com, vvsss
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