## TCS: P16 - P19

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• [Hatzipolakis - Lozada] Let ABC be a triangle, P a point and A B C the cevian triangle of P. Denote: Ab, Ac = the reflections of A in BB ,CC , resp. Bc, Ba =
Message 1 of 1 , Aug 25, 2013

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

Denote:

Ab, Ac = the reflections of A' in BB',CC', resp.

Bc, Ba = the reflections of B' in CC', AA', resp.

Ca, Cb = the reflections of C' in AA;, BB', resp.

Na,Nb,Nc = the NPC centers of ABaCa, BCbAb, CAcBc, resp.

For P = H,I the triangles ABC, NaNbNc are orthologic.

ORTHOLOGIC CENTERS. [in trilinears]

O1 stands for the orthologic center (ABC, NaNbNc)
O2 stands for the orthologic center (NaNbNc,ABC)

FOR P=I:

P16:

O1=1/(a*(a^5-(b+c)*a^4-(2*(b^2+b*c+c^2))*a^3+(2*(b+c))*(b^2-b*c+c^2)*a^2+(b^2+b*c+c^2)^2*a-(b+c)*(c^4+b^4-b*c*(2*b^2-b*c+2*c^2)))) ::

O1=(71,5341)/\(72,3585)

O1= -0.780302077498124, -0.98098196223250, 4.679945260759860

O1 on Jerabek hyperbola

P17:

O2=(2*a^7-(2*b*c+3*c^2+3*b^2)*a^5-(b+c)^3*a^4-b*c*(b^2+c^2)*a^3+2*(b^2-c^2)^2*(b+c)*a^2+(b^2+3*b*c+c^2)*(b^2-c^2)^2*a-(b-c)*(b^2-c^2)^3)/a::

O2=(1,30)/\(3,229)

O2= -0.720017732347441, -0.71365923086162, 4.467052133587395

FOR P=H:

P18:

O1=1/(a*((b^2+c^2)*a^6+(4*b^2*c^2-3*c^4-3*b^4)*a^4+(b^2+c^2)*(3*c^4-7*b^2*c^2+3*b^4)*a^2-(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2))::

No relations with ETC centers 1-5543

P19:

O2=((b^2+c^2)*a^6+(-3*b^4+8*b^2*c^2-3*c^4)*a^4+(3*(b^2+c^2))*(b^2-c^2)^2*a^2-(b^4+8*b^2*c^2+c^4)*(b^2-c^2)^2)*a::

O2=(4,51)/\(25,3357)

O2=-14.372486901377580, -16.09367038759592, 21.415891781648130

Reference:
Sun Aug 25, 2013
http://tech.groups.yahoo.com/group/Anopolis/message/877
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