Difference between revisions of "Gossard perspector"
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'''vladimir.shelomovskii@gmail.com, vvsss''' | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
+ | == Dao Thanh Oai Generalization== | ||
+ | [[File:Generalization.png|500px|right]] | ||
+ | Let line <math>\ell</math> meets sidelines <math>AB, AC, BC</math> of <math>\triangle ABC</math> at points <math>D, E, F,</math> respectively. | ||
+ | |||
+ | Let <math>Q \in \ell.</math> Let <math>K, L, M</math> be the points such that <math>QA||EM||DL, QB||KD||FM, QC||KE||FL.</math> | ||
+ | |||
+ | Similarly define points <math>P \in \ell, K_0, L_0, M_0.</math> | ||
+ | |||
+ | Let triangle <math>\triangle A'B'C'</math> be the triangle formed by the lines <math>KK_0, LL_0, MM_0.</math> | ||
+ | |||
+ | Prove that <math>\triangle A'B'C'</math> and <math>\triangle ABC</math> are homothetic and congruent, and the homothetic center lies on <math>\ell.</math> |
Revision as of 06:59, 18 January 2023
Contents
- 1 Gossard perspector X(402) and Gossard triangle
- 2 Gossard perspector of right triangle
- 3 Gossard perspector and Gossard triangle for isosceles triangle
- 4 Euler line of the triangle formed by the Euler line and the sides of a given triangle
- 5 Gossard triangle for triangle with angle 60
- 6 Gossard triangle for triangle with angle 120
- 7 Gossard perspector
- 8 Zeeman’s Generalisation
- 9 Paul Yiu's Generalization
- 10 Dao Thanh Oai Generalization
Gossard perspector X(402) and Gossard triangle
In Leonhard Euler proved that in any triangle, the orthocenter, circumcenter and centroid are collinear. We name this line the Euler line. Soon he proved that the Euler line of a given triangle together with two of its sides forms a triangle whose Euler line is in parallel with the third side of the given triangle.
Professor Harry Clinton Gossard in proved that three Euler lines of the triangles formed by the Euler line and the sides, taken by two, of a given triangle, form a triangle which is perspective with the given triangle and has the same Euler line. The center of the perspective now is known as the Gossard perspector or the Kimberling point
Let triangle be given. The Euler line crosses lines
and
at points
and
On it was found that the Gossard perspector is the centroid of the points
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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.
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Gossard perspector and Gossard triangle for isosceles triangle
It is clear that the Euler line of isosceles 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 isosceles 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
It is clear that is the midpoint of
is the midpoint
is the midpoint
with coefficient
Any isosceles triangle and its Gossard triangle are congruent.
Any isosceles triangle and its Gossard triangle have the same Euler line.
The Gossard triangle of the isosceles is the reflection of
in the Gossard perspector.
Denote
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Euler line of the triangle formed by the Euler line and the sides of a given triangle
Let the Euler line of meet the lines
and
at
and
respectively.
Euler line of the is parallel to
Similarly, Euler line of the
is parallel to
Euler line of the
is parallel to
Proof
Denote smaller angles between the Euler line and lines
and
as
and
respectively. WLOG,
It is known that
Let be circumcenter of
be Euler line of
(line).
Similarly,
Suppose,
which means
and
In this case
Similarly one can prove the claim in the other cases.
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Gossard triangle for triangle with angle 60
Let of the triangle
be
Let the Euler line of
meet the lines
and
at points
and
respectively.
Prove that
is an equilateral triangle.
Proof
Denote
It is known that
Therefore
is equilateral triangle.
Let be the triangle formed by the Euler lines of the
and the line
contains centroid
of the
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
the vertex
being the intersection of the Euler lines of the
and
We call the triangle as the Gossard triangle of the
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Gossard triangle for triangle with angle 120
Let of the triangle
be
Let the Euler line of
meet the lines
and
at points
and
respectively. Then
is an equilateral triangle.
One can prove this claim using the same formulae as in the case
Let be the triangle formed by the Euler lines of the
and the line
contains centroid
of the
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
the vertex
being the intersection of the Euler lines of the
and
We call the triangle as the Gossard triangle of the
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Gossard perspector
Let non equilateral triangle be given. The Euler line of
crosses lines
and
at points
and
respectively.
Let the point be the centroid of the set of points
Let Gossard triangle be defined as described above.
Prove that and
are homothetic and congruent, and the homothetic center is the point
the Euler line of
coincide with the Euler line of
Proof
Denote and
centroids of the triangles
and
respectively.
It is clear that
Euler line.
Let point be symmetric to the point
with respect to the point
Similarly we define points and
Similarly
and
the crosspoints of lines
and
are symmetric to the crosspoints of lines
and
therefore points
and
are symmetric to points
and
with respect to the point
is the Gossard perspector of the
It is clear that the Gossard perspector lyes on Euler line of the and
is congruent to
.
The Euler line of is symmetric to the Euler line of
with respect to
Therefore these lines coincide.
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Zeeman’s Generalisation
Let be any line parallel to the Euler line of non equilateral triangle
Let
intersect the sidelines
of
at points
respectively. Let
be the triangle formed by the Euler lines (as in previous sections) of the triangles
and
Let the point
be the centroid of the set of points
Then and
are homothetic and congruent, and the homothetic center is the point
the Euler line of
coincide with the line
and the point
is equidistant from the Euler lines.
In this case usually called the Zeeman–Gossard perspector.
One can prove this claim using the method of previous section. vladimir.shelomovskii@gmail.com, vvsss
Paul Yiu's Generalization
Let be any point in the plane of non equilateral triangle
different from its centroid
Let the line meet the sidelines
and
at
and
respectively.
Let the centroids of the triangles and
be
and
respectively.
Let be a point such that
is parallel to
and
is parallel to
Symilarly,
Let be the triangle formed by the lines
and
Let the point
be the centroid of the set of points
Then
and
are homothetic and congruent, and the homothetic center is the point
the points
and
are collinear.
One can prove this claim using the method of previous section.
The points and
are collinear.
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Dao Thanh Oai Generalization
Let line meets sidelines
of
at points
respectively.
Let Let
be the points such that
Similarly define points
Let triangle be the triangle formed by the lines
Prove that and
are homothetic and congruent, and the homothetic center lies on