Difference between revisions of "Complete Quadrilateral"
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==Newton–Gauss line== | ==Newton–Gauss line== | ||
[[File:Complete perpendicular.png|500px|right]] | [[File:Complete perpendicular.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 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>K, L,</math> and <math>N</math> be the midpoints of <math>BE, CD,</math> and <math>AF,</math> respectively. | Let points <math>K, L,</math> and <math>N</math> be the midpoints of <math>BE, CD,</math> and <math>AF,</math> respectively. | ||
+ | |||
Let points <math>H</math> and <math>H_A</math> be the orthocenters of <math>\triangle ABC</math> and <math>\triangle ADE,</math> respectively. | Let points <math>H</math> and <math>H_A</math> be the orthocenters of <math>\triangle ABC</math> and <math>\triangle ADE,</math> respectively. | ||
+ | |||
Prove that Steiner line <math>HH_A</math> is perpendicular to Gauss line <math>KLN.</math> | Prove that Steiner line <math>HH_A</math> is perpendicular to Gauss line <math>KLN.</math> | ||
<i><b>Proof</b></i> | <i><b>Proof</b></i> | ||
− | Points <math>K, L,</math> and <math>N</math> are the centers of circles with diameters <math>CD, BE,</math> and <math>AF,</math> respectively. | + | Points <math>K, L,</math> and <math>N</math> are the centers of circles with diameters <math>CD, BE,</math> and <math>AF,</math> respectively. |
+ | |||
+ | Steiner line <math>HH_A</math> is the radical axis of these circles. | ||
+ | |||
Therefore <math>HH_A \perp KL</math> as desired. | Therefore <math>HH_A \perp KL</math> as desired. | ||
+ | |||
'''vladimir.shelomovskii@gmail.com, vvsss''' | '''vladimir.shelomovskii@gmail.com, vvsss''' |
Revision as of 17:16, 9 December 2022
Complete quadrilateral
Let four lines made four triangles of a complete quadrilateral. In the diagram these are
One can see some of the properties of this configuration and their proof using the following links.
Radical axis
Let four lines made four triangles of a complete quadrilateral. In the diagram these are
Let points and
be the orthocenters of
and
respectively.
Let circles and
be the circles with diameters
and
respectively.
Prove that Steiner line
is the radical axis of
and
Proof
Let points and
be the foots of perpendiculars
and
respectively.
Denote power of point
with respect the circle
Therefore power of point
with respect these three circles is the same. These points lies on the common radical axis of
and
Steiner line
is the radical axis as desired.
vladimir.shelomovskii@gmail.com, vvsss
Newton–Gauss line
Let four lines made four triangles of a complete quadrilateral.
In the diagram these are
Let points and
be the midpoints of
and
respectively.
Let points and
be the orthocenters of
and
respectively.
Prove that Steiner line is perpendicular to Gauss line
Proof
Points and
are the centers of circles with diameters
and
respectively.
Steiner line is the radical axis of these circles.
Therefore as desired.
vladimir.shelomovskii@gmail.com, vvsss