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