Difference between revisions of "Miquel's point"
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<math>\angle MOo' = \angle MO'o' \implies</math> points <math>M, O, O', o,</math> and <math>o'</math> are concyclic as desired. | <math>\angle MOo' = \angle MO'o' \implies</math> points <math>M, O, O', o,</math> and <math>o'</math> are concyclic as desired. | ||
+ | |||
+ | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
+ | |||
+ | ==Triangle of circumcenters== | ||
+ | [[File:Miquel perspector.png|500px|right]] | ||
+ | Let four lines made four triangles of a complete quadrilateral. In the diagram these are <math>\triangle ABC, \triangle ADE, \triangle CEF, \triangle BDF.</math> | ||
+ | Let points <math>O,O_A, O_B,</math> and <math>O_C</math> be the circumcenters of <math>\triangle ABC, \triangle ADE, \triangle BDF,</math> and <math>\triangle CEF,</math> respectively. | ||
+ | Prove that <math>\triangle O_AO_BO_C \sim \triangle ABC,</math> and perspector of these triangles point <math>X</math> is the second (different from <math>M</math>) point of intersection circumcircles of <math>\triangle ABC</math> and <math>\triangle O_AO_BO_C.</math> | ||
+ | <i><b>Proof</b></i> | ||
+ | Quadrungle <math>MECF</math> is concyclic <math>\implies \angle AEM = \angle BFM \implies \angle AO_AB = 2\angle AEM = 2 \angle BFM = \angle BO_BM.</math> | ||
+ | <math>\angle CO_CM = 2\angle CFM = 2 \angle BFM = \angle BO_BM.</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> | ||
+ | Spiral similarity sentered at point <math>M</math> with rotation angle <math>\angle AMO_A = \angle BMO_B = CMO_C</math> and the coefficient of homothety <math>\frac {AM}{MO_A} = \frac {BM}{MO_B} =\frac {CM}{MO_C}</math> mapping <math>A</math> to <math>O_A</math>, <math>B</math> to <math>O_B</math>, <math>C</math> to <math>O_C \implies </math>\triangle O_AO_BO_C \sim \triangle ABC.<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>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 </math>\implies \angle O_AXO_B = \angle O_AO_CO_B = \angle ACB \implies ABCX$ is cyclic as desired. | ||
'''vladimir.shelomovskii@gmail.com, vvsss''' | '''vladimir.shelomovskii@gmail.com, vvsss''' |
Revision as of 12:19, 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 circumcircles of
and
Proof
Quadrungle
is concyclic
Spiral similarity sentered at point
with rotation angle
and the coefficient of homothety
mapping
to
,
to
,
to
\triangle O_AO_BO_C \sim \triangle ABC.$$ (Error compiling LaTeX. Unknown error_msg)\triangle AO_AM, \triangle BO_BM, \triangle CO_CM
M \implies
\implies AO_A, BO_B, CO_C
X.
\triangle AO_AM
\triangle BO_BM
O_AMO_B
O_AM
O_BM
AO_A
BO_B
\angle AXB \implies M O_AO_BX
implies M O_AO_BXO_C
\implies \angle O_AXO_B = \angle O_AO_CO_B = \angle ACB \implies ABCX$ is cyclic as desired.
vladimir.shelomovskii@gmail.com, vvsss