HYACINTHOS 22907

Antreas P. Hatzipolakis

 

[APH]

I wrote:

Let ABC be a triangle and AhBhCh, AoBoCo, AnBnCn the pedal triangles

of H,O,N resp

The midpoint of the line segment joining the orthologic centers (AhBhCh, AnBnCn) and (AoBoCo, AnBnCn) is the orthocenter of AnBnCn.

 

Which is this point (the orthocenter of the pedal triangle of N)?

 

 

 Again:

Let ABC be an acute angled triangle and T1, T2, T3 the pedal triangles
of O, H, N, resp. The triangles are pairwise orthologic.

Denote:

P13 = the orthologic center (T1, T3)
P23 = the orthologic center (T2, T3)

Conjecture:

The mdpoint of P13P23 is the orthocenter of T3.

Is it true?

In any case, is the orthocenter of the pedal triangle of N in ETC?

If no, which are its coordinates etc?

APH
............ 

 

Conjecture:

The mdpoint of P13P23 is the orthocenter of T3.

Is it true?

 

[César Lozada]:

 

Yes.

Orthocenter of T3=

    (cos(3*A)+4*sin(A)^2*cos(B-C))*(2*cos(A)*cos(B-C)+1) : : (trilinears)
= Midpoint of  (52,2888)

= Reflection of  (54/5462), (1216/1209)

= On lines  (51,195), (52,2888), (54,5462), (539,973), (546,1154), (1209,1216)

= 3*X(51)-X(195)
=  ( 9.759878873628137, -8.04113571956091, 4.703045115313558 )

 

By the way:

P13 = X(1209)

P23 = (-1+cos(2*B)+cos(2*C))*(-1+2*cos(2*A))*sec(A) : :

        = Reflection of: (54/973)

        = (4,93)/\ (6,24)

        = ( 12.481825032288840, -13.33445478195598, 7.111367777589825 )

P31=X(5)

P32=X(143)

P12=X(5)

P21=X(4)

 

Regards

César Lozada