HYACINTHOS 25498

[Antreas P. Hatzipolakis]:

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

Denote:

A", B", C" = the reflections of P in BC, CA, AB, resp.

Ma, Mb, Mc = the midpoints of AA", BB", CC", resp.

A*B*C* = the triangle bounded by A'Ma, B'Mb, C'CMc

ABC, A*B*C* are parallelogic by construction (parallelogic center (ABC, A*B*C*) = P)

(line segment A'Ma is parallel and equal to AP/2)

1. Which is the locus of the other parllelogic center (A*B*C*, ABC) as P moves on a line? The Euler line or the OI line for example.

2. Which is the locus of P such that  that A'B'C', MaMbMc are perspectine?

I lies on the locus. The perspector is the NPC center of A"B"C"

 

[César Lozada]:

1.1) For Euler line:

Locus = parabola through ETC’s 4, 5, 5640, 5663, 10095

Focus = midpoint of X(1112) and X(3154)

= (cos(2*A)+7*cos(4*A)+cos(6*A)- 15/2)*cos(B-C)+(10*cos(A)-2* cos(3*A)-2*cos(5*A))*cos(2*(B- C))+(-cos(2*A)+cos(4*A)-3/2)* cos(3*(B-C))-2*cos(5*A)+2*cos( A)-6*cos(3*A) : : (trilinears)

= 3*X(51)+X(3258) = X(476)-9*X(5640) = X(477)+7*X(9781)

= midpoint of X(1112) and X(3154)

= on lines: {30,9826}, {51,3258}, {476,1316}, {477,9781}, {523,11746}, {1112,3154}

= [ 1.079797355725402, 0.53516966478437, 2.771794395952710 ]

 

Directrix = line {526,11746}

 

ETC pairs (P, Z(A*->A)): (2,5640), (3,5), (4,4), (5,10095)

 

1.2)  For line OI:

Locus = parabola through ETCs 5, 65, 952

 

2) Locus = {Line-at-infinity} \/ {McCay cubic}

     ETC pairs (P, Z(P)=perspector(A’, Ma)) = (1,65), (3,5), (4,4),

 

César Lozada