Difference between revisions of "Gossard perspector"
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'''vladimir.shelomovskii@gmail.com, vvsss''' | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
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| + | ==Gossard perspector and Gossard triangle for equilateral triangle== | ||
| + | [[File:Gossard equilateral.png|500px|right]] | ||
| + | It is clear that the Euler line of equilateral <math>\triangle ABC (AB = AC)</math> meet the sidelines <math>BC, CA</math> and <math>AB</math> of <math>\triangle ABC</math> at <math>A'</math> and <math>A,</math> where <math>A'</math> is the midpoint of <math>BC.</math> | ||
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| + | Let <math>\triangle A'B'C'</math> be the triangle formed by the Euler lines of the <math>\triangle AA'B, \triangle AA'C,</math> and the line <math>l</math> contains <math>A</math> and parallel to <math>BC,</math> the vertex <math>B'</math> being the intersection of the Euler line of the <math>\triangle AA'C</math> and <math>l,</math> the vertex <math>C'</math> being the intersection of the Euler line of the <math>\triangle AA'B</math> and <math>l.</math> We call the triangle <math>\triangle A'B'C'</math> as the Gossard triangle of <math>\triangle ABC.</math> | ||
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| + | Let <math>\triangle ABC</math> be any equilateral triangle and let <math>\triangle A'B'C'</math> be its Gossard triangle. Then the lines <math>AA', BB',</math> and <math>CC'</math> are concurrent. We call the point of concurrence <math>Go</math> as the Gossard perspector of <math>\triangle ABC.</math> Let <math>H</math> be the orthocenter of <math>\triangle ABC, O</math> be the circumcenter of <math>\triangle ABC.</math> | ||
Revision as of 12:16, 10 January 2023
Gossard perspector X(402) and Gossard triangle
Euler proved that the Euler line of a given triangle together with two of its sides forms a triangle whose Euler line is parallel with the third side of the given triangle.
Gossard proved that the three Euler lines of the triangles formed by the Euler line and the sides, taken by twos, of a given triangle, form a triangle triply perspective with the given triangle and having the same Euler line. The orthocenters, circumcenters and centroids of these two triangles are symmetrically placed as to the center of perspective which known as Gossard perspector or Kimberling point 
Gossard perspector of right triangle
It is clear that the Euler line of right triangle 
meet the sidelines 
and 
of 
at 
and 
where is the midpoint of 
Let 
be the triangle formed by the Euler lines of the 
and the line 
contains 
and parallel to 
the vertex 
being the intersection of the Euler line of the 
and 
the vertex 
being the intersection of the Euler line of the 
and 
We call the triangle as the Gossard triangle of 
Let be any right triangle and let be its Gossard triangle. Then the lines 
and 
are concurrent. We call the point of concurrence 
as the Gossard perspector of
is the midpoint of 
is orthocenter of 
is circumcenter of 
so is midpoint of 
is the midpoint 
is the midpoint 
with coefficient 
Any right triangle and its Gossard triangle are congruent.
Any right triangle and its Gossard triangle have the same Euler line.
The Gossard triangle of the right is the reflection of in the Gossard perspector.
vladimir.shelomovskii@gmail.com, vvsss
Gossard perspector and Gossard triangle for equilateral triangle
It is clear that the Euler line of equilateral 
meet the sidelines and of at and where is the midpoint of
Let be the triangle formed by the Euler lines of the and the line contains and parallel to the vertex being the intersection of the Euler line of the and the vertex being the intersection of the Euler line of the and We call the triangle as the Gossard triangle of
Let be any equilateral triangle and let be its Gossard triangle. Then the lines and are concurrent. We call the point of concurrence as the Gossard perspector of Let 
be the orthocenter of 
be the circumcenter of
