HYACINTHOS 25769

 

[Antreas P. Hatzipolakis]:

Let ABC be an acute angled triangle.

Denote:

A', B', C' = the reflections of N in BC, CA, AB, resp.

H1, H2, H3 = the midpoints of HA', HB', HC', resp.

O1, O2, O3 = the midpoints of OA', OB', OC', resp.

The triangles H1H2H3, O1O2O3 are Eulerologic.

That is:

The Euler lines of H1O2O3, H2O3O1, H3O1O2 are concurrent at a point X

The Euler lines of O1H2H3, O2H3H1, O3H1H2 are concurrent at a point Y

The line XY is parallel to Euler line of ABC.

X, Y ?

In general triangle:

Let A*B*C* be the pedal triangle of H.

Let H* be the INCENTER of A*B*C*

(if ABC is acute angled then H* = H. If A or B or C >90 d. then H* = A or B or C, resp.)

Denote:

A', B', C' = the reflections of N in BC, CA, AB, resp.

H1, H2, H3 = the midpoints of H*A', H*B', H*C', resp.

O1, O2, O3 = the midpoints of OA', OB', OC', resp.

The triangles H1H2H3, O1O2O3 are Eulerologic.

That is:

The Euler lines of H1O2O3, H2O3O1, H3O1O2 are concurrent at a point X
The Euler lines of O1H2H3, O2H3H1, O3H1H2 are concurrent at a point Y

The line XY is parallel to OH* line.

 

[Randy Hutson]:

Hi Antreas,

Your point X:
= midpoint of X(5) and X(9730)
= complement of complement of X(568)
= X(10272) of orthocentroidal triangle
= nine-point center of triangle formed by the centroids of the three altimedial triangles
= centroid of triangle formed by the nine-point centers of the three altimedial triangles
Barycentrics: a^4(b^2 + c^2 - 6 R^2) - a^2[(b^4 + c^4) + 9(b^2 + c^2) R^2] - 3(b^2 - c^2)^2 R^2 : :
On lines {2,568}, {3,5640}, {4,7693}, {5,113}, {30,5892}, {51,549}, {52,632}, {140,143}, {381,11451}, {546,9729}, {547,6688}, {548,10110}, {631,10263}, at least
(6,9,13) values: (2.413245713025014, 1.833885769225136, 1.257245543355432)

Your point Y:
= midpoint of X(5) and X(51)
= midpoint of X(381) and X(5946)
= X(140) of orthocentroidal triangle
Barycentrics: a^2(b^4 + c^4 - a^2b^2 - a^2c^2 - 2b^2c^2)(a^4 + b^4 + c^4 - 2a^2b^2 - 2a^2c^2 - 5b^2c^2) : :
On lines {3,11451}, {4,12006}, {5,51}, {30,5892}, {140,6688}, {381,5640}, {3526,11592}, at least
(6,9,13) values: (0.336748662171200, -0.237133429977659, 3.649411320120666)

Best regards,
Randy Hutson