HYACINTHOS 24080

Antreas P. Hatzipolakis

[APH]:

Lemma:

Let ABC be a triangle, P1, P2 two points and A1B1C1,A2B2C2 the pedal triangles of P1,P2, resp.

Denote:

A12, A13 = the orthogonal projections of A1 on P1B1, P1C1, resp.

A22, A23 = the orthogonal projections of A2 on P2B2, P2C2, resp.

D1, D2 = two same points of A1A12A13, A2A22A23, resp.

The lines P1D1, P2D2 are parallel

(therefore the parallels to P1D1, P2D2 through A coincide).

Conjecture:

Let ABC be a triangle, Q a fixed point and A'B'C' the pedal triangle of Q.

Denote:

Ab, Ac = the orthogonal projection of A' on QB', QC', resp.

Bc, Ba = the orthogonal projection of B' on QC', QA', resp.

Ca, Cb = the orthogonal projection of C' on QA', QB', resp.

Pa, Pb, Pc = same points of A'AbAc, B'BcBa, C'CaCb, resp.

(ie Pa, Pb, Pc = (x:y:z) wrt the triangles A'AbAc, B'BcBa, C'CaCb)

 

The parallels to QPa, QPb, QPc through A, B, C, resp. are concurrent.

The point of concurrence is independent from Q (ie for all Q's the point of concurrence is the same, according to Lemma).

Locus:
Which is the locus of the point of concurrence as the same points Pa, Pb, Pc move on the Euler lines of A'AbAc, B'BcBa, C'CaCb, resp. ?

 

[César Lozada]:

Conjecture: Proved.

 

For P=u:v:w (trilinears),  the parallels to QPa, QPb, QPc through A, B, C, resp. are concurrent at Z(P)

 

Z(P) = 1/(a*u*cos(B)*cos(C)-(b*v+c*w) *cos(A)) : :

 

and Z(P) does not depend on Q.

 

When Pa, Pb, Pc are the same point on the Euler line, Z(P) moves on the Jerabek hyperbola.

 

ETC-mapping (P,Z(P)) (excluding circumcircle and infinity):

(1,84), (2,3426), (3,4), (4,64), (6,3424), (8,945), (20,3), (22,4846), (40,1),     

 (55,3427), (58,3429), (64,3346), (84,3345), (165,3577), (185,8884), (376,6),       

 (378,66), (548,1173), (550,54), (944,56), (1071,28), (1204,1093), (1350,2),        

 (1490,40), (1498,20), (1670,1676), (1671,1677), (1742,7350), (2071,265), (2077,80),

 (2096,1436), (2130,3183), (2131,3637), (3098,262), (3182,1490), (3183,1498),       

 (3345,3347), (3346,3348), (3347,3354), (3348,2131), (3353,3182), (3354,3473),      

 (3355,2130), (3428,7), (3430,10), (3472,3353), (3520,6145), (3522,3527),           

 (3529,3532), (3534,3431), (3576,3062), (3579,1389), (3651,65), (4297,58),          

 (4549,378), (5188,83), (5473,15), (5474,16), (5562,1105), (5732,57), (5759,55),    

 (5890,2980), (6244,1000), (6282,9), (6361,963), (6770,3440), (6773,3441), (6776,25),

(7411,1243), (7429,3657), (7464,67), (7470,695), (7488,3521), (7689,847), (7691,5),

 (8722,598), (8984,6502), (9142,8599), (9145,9141), (9409,685), (9821,3406),        

 (9841,937), (9943,961)*

 

*Numerically found. May be incomplete or contain wrong values.

 

César Lozada

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Thanks !!

> (1,84), (2,3426), (3,4), (4,64), (6,3424),,.............

The 5, where is the 5 ?  :)

APH