Difference between revisions of "Miquel's point"
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Quadrungle <math>MECF</math> is cyclic <math>\implies \angle AEM = \angle BFM \implies</math> | Quadrungle <math>MECF</math> is cyclic <math>\implies \angle AEM = \angle BFM \implies</math> | ||
− | <cmath>\angle | + | <cmath>\angle AO_AM = 2\angle AEM = 2 \angle BFM = \angle BO_BM.</cmath> |
<cmath>\angle CO_CM = 2\angle CFM = 2 \angle BFM = \angle BO_BM.</cmath> | <cmath>\angle CO_CM = 2\angle CFM = 2 \angle BFM = \angle BO_BM.</cmath> | ||
<math>AO_A = MO_A, BO_B = MO_B, CO_C = MO_C \implies \triangle AO_AM \sim \triangle BO_BM \sim \triangle CO_CM.</math> | <math>AO_A = MO_A, BO_B = MO_B, CO_C = MO_C \implies \triangle AO_AM \sim \triangle BO_BM \sim \triangle CO_CM.</math> | ||
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<math>\triangle AO_AM, \triangle BO_BM, \triangle CO_CM</math> are triangles in double perspective at point <math>M \implies</math> | <math>\triangle AO_AM, \triangle BO_BM, \triangle CO_CM</math> are triangles in double perspective at point <math>M \implies</math> | ||
− | + | These triangles are in triple perspective <math>\implies AO_A, BO_B, CO_C</math> are concurrent at the point <math>X.</math> | |
The rotation angle <math>\triangle AO_AM</math> to <math>\triangle BO_BM</math> is <math>\angle O_AMO_B</math> for sides <math>O_AM</math> and <math>O_BM</math> or angle between <math>AO_A</math> and <math>BO_B</math> which is <math>\angle AXB \implies M O_AO_BX</math> is cyclic <math>\implies M O_AO_BXO_C</math> is cyclic. | The rotation angle <math>\triangle AO_AM</math> to <math>\triangle BO_BM</math> is <math>\angle O_AMO_B</math> for sides <math>O_AM</math> and <math>O_BM</math> or angle between <math>AO_A</math> and <math>BO_B</math> which is <math>\angle AXB \implies M O_AO_BX</math> is cyclic <math>\implies M O_AO_BXO_C</math> is cyclic. | ||
− | Therefore <math> | + | Therefore <math>\angle O_AXO_B = \angle O_AO_CO_B = \angle ACB \implies ABCX</math> is cyclic as desired. |
'''vladimir.shelomovskii@gmail.com, vvsss''' | '''vladimir.shelomovskii@gmail.com, vvsss''' |
Revision as of 12:42, 6 December 2022
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.
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
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.
vladimir.shelomovskii@gmail.com, vvsss