Euler line and circle

Orthocenter properties

Let X the point with OX> = OA> + OB> + OC>
OX> - OA> = OB> + OC>. That is, M being midpoint of BC : AX> = 2 OM>, AX is then parallel to the perpendicular bisector OM of BC, hence AX is an altitude in triangle ABC, and similarily for the other vertices, hence X is really orthocenter H.

 AH> = 2 OM>   [1] 

Let A' intersect point of OA and MH. [1] ⇒ A' is center of a dilation with ratio 2 which transforms O,M into A,H hence O is midpoint of AA' ⇒ A' lies on circumcircle. And also M is midpoint of HA'.

Let H' intersect point of AH with circumcircle. AH' is perpendicular to A'H', hence A'H' parallel to BC. In triangle HH'A', as M is midpoint of HA', then K is midpoint of HH'.

 The reflection of orthocenter on a side is on the circumcircle
 The symmetric of orthocenter from midpoint of a side lies on the circumcircle. 

Euler line

Let G intersect point of OH and AM. [1] ⇒ G is the center of a dilation with ratio -2 which transforms O into H and M into A.
Hence GM = 1/3 AM, and G is then the centroid of ABC.

 The centroid, orthocenter and circumcenter are on a same line : Euler line 
 GH> = -2.GO>  [2]

Euler circle

Consider the dilation with center G and ratio -1/2.
It transforms O into midpoint ω of OH, and circumcircle into circle Γ, centered in ω, with radius R/2.
It also transforms A into M and similarily for vertices B and C transformed into midpoints of AC and AB.
Circle Γ is then the circumcircle of the median triangle (triangle of midpoints).
This dilation transforms A' into midpoint N of AH, because G is centroid of triangle AHA' : intersect point of medians AM (M midpoint of HA') and HO (=Euler line).

There is another dilation which transforms the circumcircle into circle Γ : this one transforms O into ω and has ratio +1/2. The center of this dilation is then H (HO = 2Hω, from relation [2])
By this dilation, point H' is transformed into K midpoint of HH', hence K also lies on Γ.
(note it also transforms A' into M, already seen as being on Γ)

 Midpoints of sides, feet of altitudes and midpoint of segments from orthocenter 
 to vertices all lie on a same circle, named nine points circle or Euler circle 

 Its center is midpoint of OH, its radius is R/2

There are many other known points on the Euler line or the Euler circle.
From the encyclopedia of triangle centers ETC, there are 222 triangle centers on Euler line !
Of course, center (ω midpoint of OH) of Euler circle lies on Euler line OH.
Among famous points on Euler circle, mention the Feuerbach points : touch points of Euler circle with incircle and excircles.
There are about 30 others. Most of them are transformed by the previous dilations from triangle centers lying on circumcircle.
That is considering the Euler circle as being circumcircle of the orthic triangle (triangle from the feet of altitudes), or the median trangle.


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