HYACINTHOS 25127

 

[Antreas P. Hatzipolakis]:

 

ORTHIC TRIANGLE VERSION:

 

Let ABC be a triangle and A'B'C' the pedal triangle of H..

Denote

Aa, Bb, Cc = the reflections of A, B, C in B'C', C'A', A'B', resp.

Aaa, Bbb, Ccc = the reflections of Aa,Bb,Cc in BC, CA, AB, resp.

A1, B2, C3 = the reflections of A,B,C in BC, CA, AB, resp.

A11, B22, C33 = the reflections of A1, B2, C3 in B'C', C'A', A'B', resp.

A*B*C* = the triangle bounded by AaaA11, BbbB22, CccC33

1. A'B'C', AaaBbbCcc are perspective.
Perspector (on the Euler line) ?

2. ABC, A*B*C* are parallelogic.

The parallelogic center (ABC, A*B*C*) is de Longchamps Point X20.

The other one?

EXCENTRAL TRIANGLE VERSION

Let ABC be a triangle and A'B'C' the antipedal triangle of I..

Denote

A'a, B'b,B'c = the reflections of A', B', C' in BC,CA, AB, resp.

A'aa, B'bb, C'cc = the reflections of A'a,B'b,C'c in B'C', C'A', A'B', resp.

A'1, B'2, C'3 = the reflections of A', B', C' in B'C', C'A', A'B', resp.

A'11, B'22, C'33 = the reflections of A'1, B'2, C'3 in BC, CA, AB, resp.

A*B*C* = the triangle bounded by A'aaA'11, B'bbB'22, C'ccC'33

3. ABC, A'aaB'bbC'cc are perspective.
Perspector (on the Euler line of A'B'C') ?

4. A'B'C', A*B*C* are parallelogic.

The parallelogic center (A'B'C', A*B*C*) is de Longchamps Point of A'B'C'.

The other one?

 

[César Lozada]:

ORTHIC TRIANGLE VERSION:

 

> 1. A'B'C', AaaBbbCcc are perspective. Perspector (on the Euler line) ?

X(403)

 

> 2. ABC, A*B*C* are parallelogic.

Centers: X(20) and

PC(A*->A) = (3*cos(2*A)+2*cos(4*A))*cos(B- C)-(cos(A)+2*cos(3*A))*cos(2*( B-C))+3*cos(A)+cos(3*A) : : (trilinears)

= [ -35.670462764597500, -15.80751063601123, 31.047616198191000 ]

 

EXCENTRAL TRIANGLE VERSION

> 3. ABC, A'aaB'bbC'cc are perspective. Perspector (on the Euler line of A'B'C') ?

X(36)-of-ABC = X(403)-of-A’B’C’

 

> 4. A'B'C', A*B*C* are parallelogic.

Centers: X(7991)-of-ABC=X(20)-of-A’B’C’ and

PC(A*->A’) = (8*sin(A/2)+2*sin(3*A/2))*cos( (B-C)/2)+(2*cos(A)+4)*cos(B-C) -4*sin(A/2)*cos(3*(B-C)/2)-7* cos(A)+4*cos(2*A)-3 : : (trilinears)

= On line: {1768,9669}

= [ -24.189149490581730, -30.55993777996073, 35.961767325225680 ]

 

César Lozada