Difference between revisions of "Euler line"
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Assuming that the [[nine point circle]] exists and that <math>N</math> is the center, note that a homothety centered at <math>H</math> with factor <math>2</math> brings the [[Euler point]]s <math>\{E_A, E_B, E_C\}</math> onto the circumcircle of <math>\triangle ABC</math>. Thus, it brings the nine-point circle to the circumcircle. Additionally, <math>N</math> should be sent to <math>O</math>, thus <math>N \in \overline{HO}</math> and <math>\frac{HN}{ON} = 1</math>. | Assuming that the [[nine point circle]] exists and that <math>N</math> is the center, note that a homothety centered at <math>H</math> with factor <math>2</math> brings the [[Euler point]]s <math>\{E_A, E_B, E_C\}</math> onto the circumcircle of <math>\triangle ABC</math>. Thus, it brings the nine-point circle to the circumcircle. Additionally, <math>N</math> should be sent to <math>O</math>, thus <math>N \in \overline{HO}</math> and <math>\frac{HN}{ON} = 1</math>. | ||
− | + | ==Analytic Proof of Existence== | |
− | + | Let the circumcenter be represented by the vector <math>O = (0, 0)</math>, and let vectors <math>A,B,C</math> correspond to the vertices of the triangle. It is well known the that the orthocenter is <math>H = A+B+C</math> and the centroid is <math>G = \frac{A+B+C}{3}</math>. Thus, <math>O, G, H</math> are collinear and <math>\frac{OG}{HG} = \frac{1}{2}</math> | |
[[Image:Euler Line.PNG||500px|frame|center]] | [[Image:Euler Line.PNG||500px|frame|center]] |
Revision as of 19:56, 4 August 2017
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
Given the orthic triangle of
, the Euler lines of
,
, and
concur at
, the nine-point circle of
.
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
.
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
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