Difference between revisions of "Euler line"
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+ | ==Distances along Euler line== | ||
+ | Let <math>H, G, O,</math> and <math>R</math> be orthocenter, centroid, circumcenter, and circumradius of the <math>\triangle ABC,</math> respectively. | ||
+ | |||
+ | <math>a = BC, b = AC, c = AB.</math> Prove that <math>HO^2 = R^2 (1 – 8 \cos A \cos B \cos C), GO^2 = R^2 – \frac {a^2 + b^2 + c^2}{9}.</math> | ||
+ | |||
+ | <i><b>Proof</b></i> | ||
+ | |||
+ | WLOG, <math>ABC</math> is an acute triangle, <math>\beta = \angle B > \angle C = \gamma, \alpha = \angle A.</math> | ||
+ | |||
+ | <cmath>OA = R, AH = 2 R \cos \alpha, \angle BAD = \angle OAC = 90^\circ – \beta \implies</cmath> | ||
+ | <cmath>\angle OAH = \alpha – 2\cdot (90^\circ – \beta) = \alpha + \beta + \gamma – 180^\circ + \beta – \gamma = \beta – \gamma .</cmath> | ||
+ | <cmath>HO^2 = AO^2 + AH^2 – 2 AH \cdot AO \cos \angle OAC = R^2 + (2 R \cos \alpha)^2 – 2 R (2 R \cos \alpha) \cos (\beta – \gamma).</cmath> | ||
+ | <cmath>\frac {HO^2}{R^2} = 1 + 4 \cos \alpha (\cos \alpha – \cos (\beta – \gamma) = 1 – 4 \cos \alpha (\cos (\beta + \gamma) + \cos (\ beta – \gamma) = 1 – 8 \cos \alpha \cos \ beta \cos \gamma .</cmath> | ||
+ | <cmath>\frac {HO^2}{R^2} = 1 + 4 \cos^2 \alpha – 4 \cos \alpha \cos (\beta – \gamma) = 5 - 4 \sin^2 \alpha + 4 \cos (\beta + \gamma) \cos (\beta – \gamma)</cmath> | ||
+ | <cmath>HO^2 = 5R^2 – 4R^2 \sin^2 \alpha + 2R^2 \cos 2\beta + 2R^2 \cos 2 \gamma = 9R^2 – 4R^2 \sin^2 \alpha – 4R^2 \sin^2 \beta – 4 R^2\sin^2 \gamma</cmath> | ||
+ | <cmath>HO^2 = 9R^2 – 4 R^2 \sin^2 \alpha – 4 R^2 \sin^2 \beta – 4 R^2 \sin^2 \gamma = 9R^2 – a^2 – b^2– c^2,</cmath> | ||
+ | <cmath>GO^2 = \frac {HO^2}{9} = R^2 – \frac {a^2 + b^2 + c^2}{9}.</cmath> | ||
+ | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
+ | |||
==Triangles with angles of <math>60^\circ</math> or <math>120^\circ</math>== | ==Triangles with angles of <math>60^\circ</math> or <math>120^\circ</math>== |
Revision as of 17:47, 16 November 2022
In any triangle , the Euler line is a line which passes through the orthocenter , centroid , circumcenter , nine-point center and de Longchamps point . It is named after Leonhard Euler. Its existence is a non-trivial fact of Euclidean geometry. Certain fixed orders and distance ratios hold among these points. In particular, and
Euler line is the central line .
Given the orthic triangle of , the Euler lines of ,, and concur at , the nine-point circle of .
Contents
[hide]- 1 Proof Centroid Lies on Euler Line
- 2 Another Proof
- 3 Proof Nine-Point Center Lies on Euler Line
- 4 Analytic Proof of Existence
- 5 Distances along Euler line
- 6 Triangles with angles of or
- 7 Euler lines of cyclic quadrilateral (Vittas’s theorem)
- 8 Concurrent Euler lines and Fermat points
- 9 Euler line of Gergonne triangle
- 10 Thebault point
- 11 Schiffler point
- 12 Euler line as radical axis
- 13 De Longchamps point of Euler line
- 14 See also
Proof Centroid Lies on Euler Line
This proof utilizes the concept of spiral similarity, which in this case is a rotation followed homothety. Consider the medial triangle . It is similar to . Specifically, a rotation of about the midpoint of followed by a homothety with scale factor centered at brings . Let us examine what else this transformation, which we denote as , will do.
It turns out is the orthocenter, and is the centroid of . Thus, . As a homothety preserves angles, it follows that . Finally, as it follows that Thus, are collinear, and .
Another Proof
Let be the midpoint of . Extend past to point such that . We will show is the orthocenter. Consider triangles and . Since , and they both share a vertical angle, they are similar by SAS similarity. Thus, , so lies on the altitude of . We can analogously show that also lies on the and altitudes, so is the orthocenter.
Proof Nine-Point Center Lies on Euler Line
Assuming that the nine point circle exists and that is the center, note that a homothety centered at with factor brings the Euler points onto the circumcircle of . Thus, it brings the nine-point circle to the circumcircle. Additionally, should be sent to , thus and .
Analytic Proof of Existence
Let the circumcenter be represented by the vector , and let vectors correspond to the vertices of the triangle. It is well known the that the orthocenter is and the centroid is . Thus, are collinear and
Distances along Euler line
Let and be orthocenter, centroid, circumcenter, and circumradius of the respectively.
Prove that
Proof
WLOG, is an acute triangle,
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Triangles with angles of or
Claim 1
Let the in triangle be Then the Euler line of the is parallel to the bisector of
Proof
Let be circumcircle of
Let be circumcenter of
Let be the circle symmetric to with respect to
Let be the point symmetric to with respect to
The lies on lies on
is the radius of and translation vector to is
Let be the point symmetric to with respect to Well known that lies on Therefore point lies on
Point lies on
Let be the bisector of are concurrent.
Euler line of the is parallel to the bisector of as desired.
Claim 2
Let the in triangle be Then the Euler line of the is perpendicular to the bisector of
Proof
Let be circumcircle, circumcenter, orthocenter and incenter of the points are concyclic.
The circle centered at midpoint of small arc
is rhomb.
Therefore the Euler line is perpendicular to as desired.
Claim 3
Let be a quadrilateral whose diagonals and intersect at and form an angle of If the triangles PAB, PBC, PCD, PDA are all not equilateral, then their Euler lines are pairwise parallel or coincident.
Proof
Let and be internal and external bisectors of the angle .
Then Euler lines of and are parallel to and Euler lines of and are perpendicular to as desired.
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Euler lines of cyclic quadrilateral (Vittas’s theorem)
Let be a cyclic quadrilateral with diagonals intersecting at Prove that the Euler lines of triangles are concurrent.
Proof
Let be the circumcenters (orthocenters) of triangles Let be the common bisector of and Therefore and are parallelograms with parallel sides.
bisect these angles. So points are collinear and lies on one straight line which is side of the pare vertical angles and Similarly, points are collinear and lies on another side of these angles. Similarly obtuse so points and are collinear and lies on one side and points and are collinear and lies on another side of the same vertical angles.
We use Claim and get that lines are concurrent (or parallel if or ).
Claim (Property of vertex of two parallelograms)
Let and be parallelograms, Let lines and be concurrent at point Then points and are collinear and lines and are concurrent.
Proof
We consider only the case Shift transformation allows to generalize the obtained results.
We use the coordinate system with the origin at the point and axes
We use and get points and are colinear.
We calculate point of crossing and and and and get the same result: as desired (if then point moves to infinity and lines are parallel, angles or
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Concurrent Euler lines and Fermat points
Consider a triangle with Fermat–Torricelli points and The Euler lines of the triangles with vertices chosen from and are concurrent at the centroid of triangle We denote centroids by , circumcenters by We use red color for points and lines of triangles green color for triangles and blue color for triangles
Case 1
Let be the first Fermat point of maximum angle of which smaller then Then the centroid of triangle lies on Euler line of the The pairwise angles between these Euler lines are equal
Proof
Let and be centroid, circumcenter, and circumcircle of respectevely.
Let be external for equilateral triangle is cyclic.
Point is centroid of Points and are colinear, so point lies on Euler line of
Case 2
Let be the first Fermat point of
Then the centroid of triangle lies on Euler lines of the triangles and The pairwise angles between these Euler lines are equal
Proof
Let be external for equilateral triangle, be circumcircle of is cyclic.
Point is centroid of
Points and are colinear, so point lies on Euler line of as desired.
Case 3
Let be the second Fermat point of Then the centroid of triangle lies on Euler lines of the triangles and
The pairwise angles between these Euler lines are equal
Proof
Let be internal for equilateral triangle, be circumcircle of
Let and be circumcenters of the triangles and Point is centroid of the is the Euler line of the parallel to
is bisector of is bisector of is bisector of is regular triangle.
is the inner Napoleon triangle of the is centroid of this regular triangle.
points and are collinear as desired.
Similarly, points and are collinear.
Case 4
Let and be the Fermat points of Then the centroid of point lies on Euler line is circumcenter, is centroid) of the
Proof
Step 1 We will find line which is parallel to
Let be midpoint of Let be the midpoint of
Let be point symmetrical to with respect to
as midline of
Step 2 We will prove that line is parallel to
Let be the inner Napoleon triangle. Let be the outer Napoleon triangle. These triangles are regular centered at
Points and are collinear (they lies on bisector
Points and are collinear (they lies on bisector
Points and are collinear (they lies on bisector angle between and is
Points and are concyclic Points and are concyclic
points and are concyclic
Therefore and are collinear or point lies on Euler line
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Euler line of Gergonne triangle
Prove that the Euler line of Gergonne triangle of passes through the circumcenter of triangle
Gergonne triangle is also known as the contact triangle or intouch triangle. If the inscribed circle touches the sides of at points and then is Gergonne triangle of .
Other wording: Tangents to circumcircle of are drawn at the vertices of the triangle. Prove that the circumcenter of the triangle formed by these three tangents lies on the Euler line of the original triangle.
Proof
Let and be orthocenter and circumcenter of respectively. Let be Orthic Triangle of
Then is Euler line of is the incenter of is the incenter of
Similarly,
where is the perspector of triangles and
Under homothety with center P and coefficient the incenter of maps into incenter of , circumcenter of maps into circumcenter of are collinear as desired.
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Thebault point
Let and be the altitudes of the where
a) Prove that the Euler lines of triangles are concurrent on the nine-point circle at a point T (Thebault point of )
b) Prove that if then else
Proof
Case 1 Acute triangle
a) It is known, that Euler line of acute triangle cross AB and BC (shortest and longest sides) in inner points.
Let be circumcenters of
Let and be centroids of
Denote is the circle (the nine-points circle).
is the midpoint where is the orthocenter of
Similarly
is the midline of
Let cross at point different from
spiral similarity centered at maps onto
This similarity has the rotation angle acute angle between Euler lines of these triangles is
Let these lines crossed at point Therefore points and are concyclic
Similarly, as desired.
b) Point lies on median of and divide it in ratio 2 : 1.
Point lies on Euler line of
According the Claim,
Similarly
Case 2 Obtuse triangle
a) It is known, that Euler line of obtuse cross AC and BC (middle and longest sides) in inner points.
Let be circumcenters of
Let and be centroids of
Denote is the circle (the nine-points circle).
is the midpoint where is the orthocenter of
Similarly
is the midline of
Let cross at point different from
spiral similarity centered at maps onto
This similarity has the rotation angle acute angle between Euler lines of these triangles is
Let these lines crossed at point Therefore points and are concyclic
Similarly, as desired.
b)
Point lies on median of and divide it in ratio
Point lies on Euler line of According the Claim,
Similarly
Claim (Segment crossing the median)
Let be the midpoint of side of the
Then
Proof
Let be (We use sign to denote the area of
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Schiffler point
Let and be the incenter, circumcenter, centroid, circumradius, and inradius of respectively. Then the Euler lines of the four triangles and are concurrent at Schiffler point .
Proof
We will prove that the Euler line of cross the Euler line of at such point that .
Let and be the circumcenter and centroid of respectively.
It is known that lies on circumcircle of
Denote
It is known that is midpoint point lies on median points belong the bisector of
Easy to find that ,
We use sigh [t] for area of t. We get
Using Claim we get Therefore each Euler line of triangles cross Euler line of in the same point, as desired.
Claim (Segments crossing inside triangle)
Given triangle GOY. Point lies on
Point lies on
Point lies on
Point lies on Then
Proof
Let be (We use sigh for area of vladimir.shelomovskii@gmail.com, vvsss
Euler line as radical axis
Let with altitudes and be given.
Let and be circumcircle, circumcenter, orthocenter and circumradius of respectively.
Circle centered at passes through and is tangent to the radius AO. Similarly define circles and
Then Euler line of is the radical axis of these circles.
If is acute, then these three circles intersect at two points located on the Euler line of the
Proof
The power of point with respect to and is
The power of point with respect to is
The power of point with respect to is
The power of point with respect to is
It is known that
Therefore points and lies on radical axis of these three circles as desired.
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De Longchamps point of Euler line
Definition 1
The De Longchamps’ point of a triangle is the radical center of the power circles of the triangle. Prove that De Longchamps point lies on Euler line.
We call A-power circle of a the circle centered at the midpoint point with radius The other two circles are defined symmetrically.
Proof
Let and be orthocenter, circumcenter, and De Longchamps point, respectively.
Denote power circle by power circle by WLOG,
Denote the projection of point on
We will prove that radical axes of power and power cicles is symmetric to altitude with respect Further, we will conclude that the point of intersection of the radical axes, symmetrical to the heights with respect to O, is symmetrical to the point of intersection of the heights with respect to
Point is the crosspoint of the center line of the power and power circles and there radical axis. We use claim and get:
and are the medians, so
We use Claim some times and get: radical axes of power and power cicles is symmetric to altitude with respect
Similarly radical axes of power and power cicles is symmetric to altitude radical axes of power and power cicles is symmetric to altitude with respect Therefore the point of intersection of the radical axes, symmetrical to the heights with respect to is symmetrical to the point of intersection of the heights with respect to lies on Euler line of
Claim (Distance between projections)
Definition 2
We call circle of a the circle centered at with radius The other two circles are defined symmetrically. The De Longchamps point of a triangle is the radical center of circle, circle, and circle of the triangle (Casey – 1886). Prove that De Longchamps point under this definition is the same as point under Definition 1.
Proof
Let and be orthocenter, centroid, and De Longchamps point, respectively. Let cross at points and The other points are defined symmetrically. Similarly is diameter
Therefore is anticomplementary triangle of is orthic triangle of So is orthocenter of
as desired.
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See also
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