HYACINTHOS 25511

[Antreas P. Hatzipolakis]:

Let ABC be a triangle, P a point and PaPbPc the pedal triangle of P.

Denote:

A'B'C' = the midway triangle of O
(ie A', B', C' = midpoints of OA, OB, OC, resp.)

A"B"C" = the tangential triangle of A'B'C'.
(ie the antipedal triangle of O wrt triangle A'B'C')

Ma, Mb, Mc = the midpoints of APa, BPb, CPc, resp

R1 = the radical axis of the circles (Mb, MbB'), (Mc, McC')
R2 = the radical axis of the circles (Mc, McC'), (Ma, MaA')
R3 = the radical axis of the circles (Ma, MaA'), (Mb, MaB')
[the radical center of the circles is the point they concur at = the NPC center of ABC]

Ra, Rb, Rc = the reflections of R1, R2, R3 in OA", OB", OC", resp.
[OA", OB", OC' are the perpendicular bisectors of ABC and A'B'C']

A*B*C* = the triangle bounded by Ra, Rb, Rc.

1. Which is the locus of P such that the triangles A"B"C" and A*B*C* are parallelogic?

H lies on the locus. The parallelogic center (A"B"C", A*B*C*) lies on the Euler line of ABC.

O lies on the locus. Ra, Rb, Rc are concurrent on the Euler line at the reflection of N in O.
(R1, R2, R3 are parallels to OA",OB",OC", resp.)

2. Which is the locus of P such that Ra, Rb, Rc are concurrent?
O lies on the locus.

 

[Angel Montesdeoca]:

**** The locus of P such that the triangles A"B"C" and A*B*C* are parallelogic is the Euler line and the circumcircle.

== Parallelogic center (A"B"C", A*B*C*)
 
-- P lies on Euler line:
If P = O + t H then the parallelogic center (A"B"C", A*B*C*) is  Q = (OH^2 t + R^2) O + t R^2  H (lies on Euler line).

In particular, Q=O if t=0,  ie. P=O
                     Q=H if t=-R^2/OH^2,  ie. P = OH^2 O - R^2 H =  X(11250) =
         (a^2 (a^8-2 a^6 (b^2+c^2)+6 a^4 b^2 c^2+a^2 (2 b^6-3 b^4 c^2-3 b^2 c^4+2 c^6)-(b^2-c^2)^2 (b^4+3 b^2 c^2+c^4)):...:...).
                     Q=N if  t=R^2/(R^2-OH^2),  ie. P = (R^2-OH^2)  O +R^2 H = X(3520).
                    
Other pairs {P,Q}:  {20,1658}, {30,186}, {376,7502}, {378,6644}, {548,7512}, {550,7488}, {3516,6642},  {3522,7525}, {7464,7575}, {7502,10298}, {8703,6636}, {10226,3520}, {11410,9818}, {11413,26}

-- If P lies on circumcircle  then the parallelogic center (A"B"C", A*B*C*) is  the homothetic of P under h(X(186), (a^2 b^2 c^2)/(8SA SB SC)) (lies on circumcircle of A"B"C", of center X(1658) and passes through X(5961)).


== Parallelogic center (A*B*C*, A"B"C")
 
-- P lies on Euler line then the parallelogic center (A*B*C*,A"B"C") lies on the line X(550)X(1511)
Pairs {P,parallelogic center (A*B*C*,A"B"C")} :  {3,550},  {20,10282}, {30,1511},  {? , 2777}, {? ,2883}

-- If P lies on circumcircle  then the parallelogic center  (A*B*C*,A"B"C") lies on the conic

(a^6 b^4 c^2+a^4 b^6 c^2-5 a^2 b^8 c^2+3 b^10 c^2+a^6 b^2 c^4-6 a^4 b^4 c^4+5 a^2 b^6 c^4-12 b^8 c^4+a^4 b^2 c^6+5 a^2 b^4 c^6+18 b^6 c^6-5 a^2 b^2 c^8-12 b^4 c^8+3 b^2 c^10) x^2 + (-2 a^12+7 a^10 b^2-8 a^8 b^4+2 a^6 b^6+2 a^4 b^8-a^2 b^10+7 a^10 c^2-18 a^8 b^2 c^2+22 a^6 b^4 c^2-18 a^4 b^6 c^2+7 a^2 b^8 c^2-8 a^8 c^4+22 a^6 b^2 c^4+8 a^4 b^4 c^4-6 a^2 b^6 c^4+2 a^6 c^6-18 a^4 b^2 c^6-6 a^2 b^4 c^6+2 a^4 c^8+7 a^2 b^2 c^8-a^2 c^10) y z + ... = 0,

of center X(10282) = 5 X(3) - X(64)
(see Hyacinthos 24665 ) and passes through X(3) and X(6759).
Pairs {P,parallelogic center (A*B*C*,A"B"C")} : {74,6759}, {110,3} 



Angel Montesdeoca