Difference between revisions of "Simson line"
(→Simson line (main)) |
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==Simson line (main)== | ==Simson line (main)== | ||
[[File:Simson line.png|300px|right]] | [[File:Simson line.png|300px|right]] | ||
− | Let a triangle <math>\triangle ABC</math> and a point <math>P</math> be given. Let <math>D, E,</math> and <math>F</math> be the foots of the perpendiculars dropped from P to lines AB, AC, and BC, respectively. | + | [[File:Simson line inverse.png|300px|right]] |
+ | Let a triangle <math>\triangle ABC</math> and a point <math>P</math> be given. | ||
+ | |||
+ | Let <math>D, E,</math> and <math>F</math> be the foots of the perpendiculars dropped from P to lines AB, AC, and BC, respectively. | ||
Then points <math>D, E,</math> and <math>F</math> are collinear iff the point <math>P</math> lies on circumcircle of <math>\triangle ABC.</math> | Then points <math>D, E,</math> and <math>F</math> are collinear iff the point <math>P</math> lies on circumcircle of <math>\triangle ABC.</math> | ||
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Let the points <math>D, E,</math> and <math>F</math> be collinear. | Let the points <math>D, E,</math> and <math>F</math> be collinear. | ||
− | + | ||
<math>AEPD</math> is cyclic <math>\implies \angle APE = \angle ADE, \angle APE = \angle BAC.</math> | <math>AEPD</math> is cyclic <math>\implies \angle APE = \angle ADE, \angle APE = \angle BAC.</math> | ||
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<math>ACBP</math> is cyclis as desired. | <math>ACBP</math> is cyclis as desired. | ||
+ | |||
+ | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
+ | |||
+ | ==Problem== | ||
+ | [[File:Problem on Simson line.png |400px|right]] | ||
+ | |||
+ | Let the points <math>A, B,</math> and <math>C</math> be collinear and the point <math>P \notin AB.</math> | ||
+ | |||
+ | Let <math>O,O_0,</math> and <math>O_1</math> be the circumcenters of triangles <math>\triangle ABP, \triangle ACP,</math> and <math>\triangle BCP.</math> | ||
+ | |||
+ | Prove that <math>P</math> lies on circumcircle of <math>\triangle OO_0O_1.</math> | ||
+ | |||
+ | <i><b>Proof</b></i> | ||
+ | |||
+ | Let <math>D, E,</math> and <math>F</math> be the midpoints of segments <math>AB, AC,</math> and <math>BC,</math> respectively. | ||
+ | |||
+ | Then points <math>D, E,</math> and <math>F</math> are collinear <math>(DE||AB, EF||DC).</math> | ||
+ | |||
+ | <math>PD \perp OO_0, PE \perp OO_1, PF \perp O_0O_1 \implies</math> | ||
+ | <math>DEF</math> is Simson line of <math>\triangle OO_0O_1 \implies P</math> lies on circumcircle of <math>\triangle OO_0O_1</math> as desired. | ||
'''vladimir.shelomovskii@gmail.com, vvsss''' | '''vladimir.shelomovskii@gmail.com, vvsss''' |
Revision as of 15:55, 30 November 2022
In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear.
Proof
In the shown diagram, we draw additional lines and
. Then, we have cyclic quadrilaterals
,
, and
. (more will be added)
Simson line (main)
Let a triangle and a point
be given.
Let and
be the foots of the perpendiculars dropped from P to lines AB, AC, and BC, respectively.
Then points and
are collinear iff the point
lies on circumcircle of
Proof
Let the point be on the circumcircle of
is cyclic
is cyclic
is cyclic
and
are collinear as desired.
Proof
Let the points and
be collinear.
is cyclic
is cyclic
is cyclis as desired.
vladimir.shelomovskii@gmail.com, vvsss
Problem
Let the points and
be collinear and the point
Let and
be the circumcenters of triangles
and
Prove that lies on circumcircle of
Proof
Let and
be the midpoints of segments
and
respectively.
Then points and
are collinear
is Simson line of
lies on circumcircle of
as desired.
vladimir.shelomovskii@gmail.com, vvsss