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Difference between revisions of "Euler line"

(→‎Concurrent Euler lines and Fermat points)
(→‎Concurrent Euler lines and Fermat points)
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Symilarly, <math>M</math> lies on Euler lines of <math>\triangle BCF</math> and <math>\triangle ACF</math> as desired.
 
Symilarly, <math>M</math> lies on Euler lines of <math>\triangle BCF</math> and <math>\triangle ACF</math> as desired.
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<i><b>Case 2</b></i>
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[[File:Fermat 1 130 lines.png|500px|right]]
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Let <math>F</math> be Fermat point <math>F_1</math> of <math>\triangle ABC,  \angle BAC > 120^\circ.</math> Then the centroid of triangle <math>ABC</math> lies on Euler line of the <math>\triangle ABF.</math>
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<i><b>Proof</b></i>
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Let <math>\triangle ABD</math> be external for <math>\triangle ABC</math> equilateral triangle, <math>\omega</math> be circumcircle of <math>\triangle ABD \implies F = CD \cap \omega.</math>
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Let <math>M_0</math> be the centroid of <math>\triangle ADF, \vec M_0 = \frac {\vec A + \vec F + \vec D}{3}.</math>
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<math>\angle AFB = \angle BFD = 60^\circ,\angle AFD = 120^\circ \implies</math>
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Euler line <math>O_1M_0</math> of <math>\triangle ADF</math> is parallel to <math>BF.</math>
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Point <math>O_1</math> is the circumcenter of <math>\triangle ABF, \triangle ADF, \triangle ABD</math> and centroid of the equilateral <math>\triangle ABD</math>
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<cmath>\implies \vec O_1 = \frac {\vec A + \vec B + \vec D}{3}.</cmath>
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Therefore <math>\vec {O_1 M_0} = \frac {\vec A + \vec F + \vec D}{3} - \frac {\vec A + \vec B + \vec D}{3} = \frac {\vec F - \vec B}{3},</math>
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<cmath>\vec {O_1 M_1} = \frac {\vec A + \vec F + \vec B}{3} - \frac {\vec A + \vec B + \vec D}{3} = \frac {\vec F - \vec D}{3} \implies \angle M_0O_1M_1 = \angle BFD =  60^\circ.</cmath>
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<cmath>O_1M_0 || BF, \angle M_0O_1M_1 = \angle BFD \implies O_1M_1 || CD.</cmath>
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<cmath>\vec M = \frac {\vec A +\vec B + \vec C}{3},\vec  M_1 = \frac {\vec A + \vec B + \vec F}{3} \implies \vec {M_1M} =  \frac {\vec {FC}}{3}.</cmath>
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Point <math>M_1 </math> lies on Euler line of <math>\triangle ABF</math>, this line is parallel to <math>FC, \implies M \in O_1M_1</math> as desired.
  
 
==See also==
 
==See also==

Revision as of 03:16, 20 October 2022

In any triangle $\triangle ABC$, the Euler line is a line which passes through the orthocenter $H$, centroid $G$, circumcenter $O$, nine-point center $N$ and de Longchamps point $L$. 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, $\overline{OGNH}$ and $OG:GN:NH = 2:1:3$


Euler line is the central line $L_{647}$.


Given the orthic triangle $\triangle H_AH_BH_C$ of $\triangle ABC$, the Euler lines of $\triangle AH_BH_C$,$\triangle BH_CH_A$, and $\triangle CH_AH_B$ concur at $N$, the nine-point circle of $\triangle ABC$.

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 $\triangle O_AO_BO_C$. It is similar to $\triangle ABC$. Specifically, a rotation of $180^\circ$ about the midpoint of $O_BO_C$ followed by a homothety with scale factor $2$ centered at $A$ brings $\triangle ABC \to \triangle O_AO_BO_C$. Let us examine what else this transformation, which we denote as $\mathcal{S}$, will do.

It turns out $O$ is the orthocenter, and $G$ is the centroid of $\triangle O_AO_BO_C$. Thus, $\mathcal{S}(\{O_A, O, G\}) = \{A, H, G\}$. As a homothety preserves angles, it follows that $\measuredangle O_AOG = \measuredangle AHG$. Finally, as $\overline{AH} || \overline{O_AO}$ it follows that \[\triangle AHG = \triangle O_AOG\] Thus, $O, G, H$ are collinear, and $\frac{OG}{HG} = \frac{1}{2}$.

Another Proof

Let $M$ be the midpoint of $BC$. Extend $CG$ past $G$ to point $H'$ such that $CG = \frac{1}{2} GH$. We will show $H'$ is the orthocenter. Consider triangles $MGO$ and $AGH'$. Since $\frac{MG}{GA}=\frac{H'G}{GC} = \frac{1}{2}$, and they both share a vertical angle, they are similar by SAS similarity. Thus, $AH' \parallel OM \perp BC$, so $H'$ lies on the $A$ altitude of $\triangle ABC$. We can analogously show that $H'$ also lies on the $B$ and $C$ altitudes, so $H'$ is the orthocenter.

Proof Nine-Point Center Lies on Euler Line

Assuming that the nine point circle exists and that $N$ is the center, note that a homothety centered at $H$ with factor $2$ brings the Euler points $\{E_A, E_B, E_C\}$ onto the circumcircle of $\triangle ABC$. Thus, it brings the nine-point circle to the circumcircle. Additionally, $N$ should be sent to $O$, thus $N \in \overline{HO}$ and $\frac{HN}{ON} = 1$.

Analytic Proof of Existence

Let the circumcenter be represented by the vector $O = (0, 0)$, and let vectors $A,B,C$ correspond to the vertices of the triangle. It is well known the that the orthocenter is $H = A+B+C$ and the centroid is $G = \frac{A+B+C}{3}$. Thus, $O, G, H$ are collinear and $\frac{OG}{HG} = \frac{1}{2}$

Euler Line.PNG

Euler line for a triangle with an angle of 120$^\circ.$

120 orthocenter.png

Let the $\angle C$ in triangle $ABC$ be $120^\circ.$ Then the Euler line of the $\triangle ABC$ is parallel to the bisector of $\angle C.$

Proof

Let $\omega$ be circumcircle of $\triangle ABC.$

Let $O$ be circumcenter of $\triangle ABC.$

Let $\omega'$ be the circle symmetric to $\omega$ with respect to $AB.$

Let $E$ be the point symmetric to $O$ with respect to $AB.$

The $\angle C = 120^\circ \implies O$ lies on $\omega', E$ lies on $\omega.$

$EO$ is the radius of $\omega$ and $\omega' \implies$ translation vector $\omega'$ to $\omega$ is $\vec {EO}.$

Let $H'$ be the point symmetric to $H$ with respect to $AB.$ Well known that $H'$ lies on $\omega.$ Therefore point $H$ lies on $\omega'.$

Point $C$ lies on $\omega, CH || OE \implies CH = OE.$

Let $CD$ be the bisector of $\angle C \implies E,O,D$ are concurrent. $OD = HC, OD||HC \implies CD || HO \implies$

Euler line $HO$ of the $\triangle ABC$ is parallel to the bisector $CD$ of $\angle C$ as desired.

vladimir.shelomovskii@gmail.com, vvsss

Concurrent Euler lines and Fermat points

Fermat point lines.png

Consider a triangle $ABC$ with Fermat–Torricelli points $F_1$ and $F_2.$ The Euler lines of the $10$ triangles with vertices chosen from $A, B, C, F_1,$ and $F_2$ are concurrent at the centroid of triangle $ABC.$

Case 1

Let $F$ be Fermat point $F_1$ of $\triangle ABC$ maximum angle of which smaller then $120^\circ.$ Then the centroid of triangle $ABC$ lies on Euler line of the $\triangle ABF.$

Proof

$\angle AFB = 120^\circ, CF$ is bisector $\angle AFB.$

As shown above, the Euler line $O_1H_1$ of the $\triangle ABF$ is parallel to $CF.$

Let $H, M,$ and $O$ be orthocenter, centroid and circumcenter of $\triangle ABC,$ respectively. Let $M_1$ be centroid of $\triangle ABF.$ \[\vec M = \frac {\vec A + \vec B + \vec C}{3}, \vec M_1 = \frac {\vec A + \vec B + \vec F}{3} \implies \vec {M_1M} = \frac {\vec {FC}}{3}.\] Point $M_1$ lies on Euler line of $\triangle ABF$, this line is parallel to $FC \implies M \in O_1 H_1.$

Symilarly, $M$ lies on Euler lines of $\triangle BCF$ and $\triangle ACF$ as desired.

Case 2

Fermat 1 130 lines.png

Let $F$ be Fermat point $F_1$ of $\triangle ABC, \angle BAC > 120^\circ.$ Then the centroid of triangle $ABC$ lies on Euler line of the $\triangle ABF.$

Proof

Let $\triangle ABD$ be external for $\triangle ABC$ equilateral triangle, $\omega$ be circumcircle of $\triangle ABD \implies F = CD \cap \omega.$

Let $M_0$ be the centroid of $\triangle ADF, \vec M_0 = \frac {\vec A + \vec F + \vec D}{3}.$ $\angle AFB = \angle BFD = 60^\circ,\angle AFD = 120^\circ \implies$

Euler line $O_1M_0$ of $\triangle ADF$ is parallel to $BF.$

Point $O_1$ is the circumcenter of $\triangle ABF, \triangle ADF, \triangle ABD$ and centroid of the equilateral $\triangle ABD$ \[\implies \vec O_1 = \frac {\vec A + \vec B + \vec D}{3}.\] Therefore $\vec {O_1 M_0} = \frac {\vec A + \vec F + \vec D}{3} - \frac {\vec A + \vec B + \vec D}{3} = \frac {\vec F - \vec B}{3},$

\[\vec {O_1 M_1} = \frac {\vec A + \vec F + \vec B}{3} - \frac {\vec A + \vec B + \vec D}{3} = \frac {\vec F - \vec D}{3} \implies \angle M_0O_1M_1 = \angle BFD = 60^\circ.\] \[O_1M_0 || BF, \angle M_0O_1M_1 = \angle BFD \implies O_1M_1 || CD.\] \[\vec M = \frac {\vec A +\vec B + \vec C}{3},\vec M_1 = \frac {\vec A + \vec B + \vec F}{3} \implies \vec {M_1M} = \frac {\vec {FC}}{3}.\] Point $M_1$ lies on Euler line of $\triangle ABF$, this line is parallel to $FC, \implies M \in O_1M_1$ as desired.

See also

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