Difference between revisions of "Gauss line"
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==Existence of the Gauss line== | ==Existence of the Gauss line== | ||
− | The complete quadilateral <math>ABCDEF (E = AD \cap BC, F = AB \cap CD)</math> be given. | + | [[File:Gauss line 1.png|400px|right]] |
− | Denote <math>O | + | The complete quadilateral <math>ABCDEF</math> <math>(E = AD \cap BC, F = AB \cap CD)</math> be given. |
+ | Denote <math>O, O_1, O_2</math> the midpoints of <math>AC, BD, EF,</math> respectively. | ||
− | + | Denote <math>H, H_1, H_2, H_3</math> the orthocenters of the <math>\triangle CDE, \triangle BCF, \triangle ABE, \triangle ADF,</math> respectively. | |
− | a) points <math>O, O_1, O_2</math> are collinear; | + | Denote <math>\omega, \Omega, \theta, \alpha,</math> and <math>\beta</math> the circles with diameters <math>AC, BD, EF, CD,</math> and <math>CE,</math> respectively. |
+ | |||
+ | Prove a) points <math>O, O_1, O_2</math> are collinear; | ||
b) <math>OO_1 \perp HH_1;</math> | b) <math>OO_1 \perp HH_1;</math> | ||
− | + | ||
c) points <math>H, H_1, H_2, H_3</math> are collinear. | c) points <math>H, H_1, H_2, H_3</math> are collinear. | ||
<i><b>Proof</b></i> | <i><b>Proof</b></i> | ||
− | Let <cmath>C_1 \in AD, CC_1 \perp | + | Let <cmath>C_1 \in AD, CC_1 \perp AD, D_1 \in BC, DD_1 \perp BC, E_1 \in CD, EE_1 \perp CD \implies</cmath> |
− | <cmath>H = CC_1 \cap DD_1 \cap EE_1, C_1 \in \alpha, D_1 \in \alpha, C_1 \in \beta, E_1 \in \ | + | <cmath>H = CC_1 \cap DD_1 \cap EE_1, C_1 \in \alpha, D_1 \in \alpha, C_1 \in \beta, E_1 \in \beta \implies</cmath> |
<math>H</math> is the radical center of <math>\omega, \Omega,</math> and <math>\alpha \implies H</math> lies on the radical axes of <math>\omega</math> and <math>\Omega.</math> | <math>H</math> is the radical center of <math>\omega, \Omega,</math> and <math>\alpha \implies H</math> lies on the radical axes of <math>\omega</math> and <math>\Omega.</math> | ||
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Similarly, using circles with diameters <math>BC</math> and <math>FC</math> one can prove that <math>H_1</math> lies on the radical axes of <math>\omega</math> and <math>\Omega</math> and on the radical axes of <math>\omega</math> and <math>\theta.</math> | Similarly, using circles with diameters <math>BC</math> and <math>FC</math> one can prove that <math>H_1</math> lies on the radical axes of <math>\omega</math> and <math>\Omega</math> and on the radical axes of <math>\omega</math> and <math>\theta.</math> | ||
− | Therefore <math>HH_1 \perp OO_1, HH_1 \perp OO_2 \implies</math> points <math>O, O_1,</math> and <math>O_2</math> are collinear | + | Therefore <math>HH_1 \perp OO_1, HH_1 \perp OO_2 \implies</math> points <math>O, O_1,</math> and <math>O_2</math> are collinear. |
− | Similarly, one can prove that <math>H_2</math> and <math>H_3</math> lie on the radical axes of <math>\omega</math> and <math>\Omega \implies </math> points H, H_1, H_2 | + | |
+ | It is clear that <math>HH_1</math> is the perpendicular to the line <math>OO_1O_2.</math>. | ||
+ | |||
+ | Similarly, one can prove that <math>H_2</math> and <math>H_3</math> lie on the radical axes of <math>\omega</math> and <math>\Omega \implies </math> points <math>H, H_1, H_2</math> and <math>H_3</math> are collinear. | ||
+ | *[[Steiner line]] | ||
+ | '''vladimir.shelomovskii@gmail.com, vvsss''' |
Latest revision as of 06:35, 26 April 2023
The Gauss line (or Gauss–Newton line) is the line joining the midpoints of the three diagonals of a complete quadrilateral.
Existence of the Gauss line
The complete quadilateral
be given.
Denote
the midpoints of
respectively.
Denote the orthocenters of the
respectively.
Denote and
the circles with diameters
and
respectively.
Prove a) points are collinear;
b)
c) points are collinear.
Proof
Let
is the radical center of
and
lies on the radical axes of
and
is the radical center of
and
lies on the radical axes of
and
Similarly, using circles with diameters and
one can prove that
lies on the radical axes of
and
and on the radical axes of
and
Therefore points
and
are collinear.
It is clear that is the perpendicular to the line
.
Similarly, one can prove that and
lie on the radical axes of
and
points
and
are collinear.
vladimir.shelomovskii@gmail.com, vvsss