Παρασκευή, 16 Μαρτίου 2012

Perspective


Let ABC be a triangle, A'B'C' the medial triangle and A"B"C" the circumcevian triangle with respect A'B'C'.


The circles with diameters HA",HB",HC" intersect the NPC again at A1,B1,C1, resp.

The triangles ABC, A1B1C1 are perspective.

Perspector?

APH, 16 March 2012

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Not in ETC. It is the isotomic conjugate of

(SA (b^4 + c^4-a^4), SB (a^4 - b^4 + c^4), SC (a^4 + b^4 - c^4))

The locus of P, instead of H, is trilinear polar of X648 (containing H) + NPC + a cubic:

5 a^12 x^3 - 28 a^10 b^2 x^3 + 53 a^8 b^4 x^3 - 40 a^6 b^6 x^3 +
7 a^4 b^8 x^3 + 4 a^2 b^10 x^3 - b^12 x^3 - 28 a^10 c^2 x^3 +
110 a^8 b^2 c^2 x^3 - 120 a^6 b^4 c^2 x^3 + 36 a^4 b^6 c^2 x^3 +
4 a^2 b^8 c^2 x^3 - 2 b^10 c^2 x^3 + 53 a^8 c^4 x^3 -
120 a^6 b^2 c^4 x^3 + 42 a^4 b^4 c^4 x^3 - 8 a^2 b^6 c^4 x^3 +
b^8 c^4 x^3 - 40 a^6 c^6 x^3 + 36 a^4 b^2 c^6 x^3 -
8 a^2 b^4 c^6 x^3 + 4 b^6 c^6 x^3 + 7 a^4 c^8 x^3 +
4 a^2 b^2 c^8 x^3 + b^4 c^8 x^3 + 4 a^2 c^10 x^3 - 2 b^2 c^10 x^3 -
c^12 x^3 + a^12 x^2 y - 4 a^10 b^2 x^2 y - 7 a^8 b^4 x^2 y +
40 a^6 b^6 x^2 y - 53 a^4 b^8 x^2 y + 28 a^2 b^10 x^2 y -
5 b^12 x^2 y - 2 a^10 c^2 x^2 y + 24 a^8 b^2 c^2 x^2 y -
44 a^6 b^4 c^2 x^2 y + 48 a^4 b^6 c^2 x^2 y - 34 a^2 b^8 c^2 x^2 y +
8 b^10 c^2 x^2 y - 17 a^8 c^4 x^2 y + 46 a^4 b^4 c^4 x^2 y +
3 b^8 c^4 x^2 y + 36 a^6 c^6 x^2 y - 24 a^4 b^2 c^6 x^2 y +
4 a^2 b^4 c^6 x^2 y - 8 b^6 c^6 x^2 y - 17 a^4 c^8 x^2 y +
4 a^2 b^2 c^8 x^2 y + b^4 c^8 x^2 y - 2 a^2 c^10 x^2 y +
c^12 x^2 y - 5 a^12 x y^2 + 28 a^10 b^2 x y^2 - 53 a^8 b^4 x y^2 +
40 a^6 b^6 x y^2 - 7 a^4 b^8 x y^2 - 4 a^2 b^10 x y^2 + b^12 x y^2 +
8 a^10 c^2 x y^2 - 34 a^8 b^2 c^2 x y^2 + 48 a^6 b^4 c^2 x y^2 -
44 a^4 b^6 c^2 x y^2 + 24 a^2 b^8 c^2 x y^2 - 2 b^10 c^2 x y^2 +
3 a^8 c^4 x y^2 + 46 a^4 b^4 c^4 x y^2 - 17 b^8 c^4 x y^2 -
8 a^6 c^6 x y^2 + 4 a^4 b^2 c^6 x y^2 - 24 a^2 b^4 c^6 x y^2 +
36 b^6 c^6 x y^2 + a^4 c^8 x y^2 + 4 a^2 b^2 c^8 x y^2 -
17 b^4 c^8 x y^2 - 2 b^2 c^10 x y^2 + c^12 x y^2 - a^12 y^3 +
4 a^10 b^2 y^3 + 7 a^8 b^4 y^3 - 40 a^6 b^6 y^3 + 53 a^4 b^8 y^3 -
28 a^2 b^10 y^3 + 5 b^12 y^3 - 2 a^10 c^2 y^3 + 4 a^8 b^2 c^2 y^3 +
36 a^6 b^4 c^2 y^3 - 120 a^4 b^6 c^2 y^3 + 110 a^2 b^8 c^2 y^3 -
28 b^10 c^2 y^3 + a^8 c^4 y^3 - 8 a^6 b^2 c^4 y^3 +
42 a^4 b^4 c^4 y^3 - 120 a^2 b^6 c^4 y^3 + 53 b^8 c^4 y^3 +
4 a^6 c^6 y^3 - 8 a^4 b^2 c^6 y^3 + 36 a^2 b^4 c^6 y^3 -
40 b^6 c^6 y^3 + a^4 c^8 y^3 + 4 a^2 b^2 c^8 y^3 + 7 b^4 c^8 y^3 -
2 a^2 c^10 y^3 + 4 b^2 c^10 y^3 - c^12 y^3 + a^12 x^2 z -
2 a^10 b^2 x^2 z - 17 a^8 b^4 x^2 z + 36 a^6 b^6 x^2 z -
17 a^4 b^8 x^2 z - 2 a^2 b^10 x^2 z + b^12 x^2 z -
4 a^10 c^2 x^2 z + 24 a^8 b^2 c^2 x^2 z - 24 a^4 b^6 c^2 x^2 z +
4 a^2 b^8 c^2 x^2 z - 7 a^8 c^4 x^2 z - 44 a^6 b^2 c^4 x^2 z +
46 a^4 b^4 c^4 x^2 z + 4 a^2 b^6 c^4 x^2 z + b^8 c^4 x^2 z +
40 a^6 c^6 x^2 z + 48 a^4 b^2 c^6 x^2 z - 8 b^6 c^6 x^2 z -
53 a^4 c^8 x^2 z - 34 a^2 b^2 c^8 x^2 z + 3 b^4 c^8 x^2 z +
28 a^2 c^10 x^2 z + 8 b^2 c^10 x^2 z - 5 c^12 x^2 z +
10 a^12 x y z - 36 a^10 b^2 x y z + 22 a^8 b^4 x y z +
8 a^6 b^6 x y z + 22 a^4 b^8 x y z - 36 a^2 b^10 x y z +
10 b^12 x y z - 36 a^10 c^2 x y z + 124 a^8 b^2 c^2 x y z -
88 a^6 b^4 c^2 x y z - 88 a^4 b^6 c^2 x y z +
124 a^2 b^8 c^2 x y z - 36 b^10 c^2 x y z + 22 a^8 c^4 x y z -
88 a^6 b^2 c^4 x y z + 132 a^4 b^4 c^4 x y z -
88 a^2 b^6 c^4 x y z + 22 b^8 c^4 x y z + 8 a^6 c^6 x y z -
88 a^4 b^2 c^6 x y z - 88 a^2 b^4 c^6 x y z + 8 b^6 c^6 x y z +
22 a^4 c^8 x y z + 124 a^2 b^2 c^8 x y z + 22 b^4 c^8 x y z -
36 a^2 c^10 x y z - 36 b^2 c^10 x y z + 10 c^12 x y z + a^12 y^2 z -
2 a^10 b^2 y^2 z - 17 a^8 b^4 y^2 z + 36 a^6 b^6 y^2 z -
17 a^4 b^8 y^2 z - 2 a^2 b^10 y^2 z + b^12 y^2 z +
4 a^8 b^2 c^2 y^2 z - 24 a^6 b^4 c^2 y^2 z + 24 a^2 b^8 c^2 y^2 z -
4 b^10 c^2 y^2 z + a^8 c^4 y^2 z + 4 a^6 b^2 c^4 y^2 z +
46 a^4 b^4 c^4 y^2 z - 44 a^2 b^6 c^4 y^2 z - 7 b^8 c^4 y^2 z -
8 a^6 c^6 y^2 z + 48 a^2 b^4 c^6 y^2 z + 40 b^6 c^6 y^2 z +
3 a^4 c^8 y^2 z - 34 a^2 b^2 c^8 y^2 z - 53 b^4 c^8 y^2 z +
8 a^2 c^10 y^2 z + 28 b^2 c^10 y^2 z - 5 c^12 y^2 z - 5 a^12 x z^2 +
8 a^10 b^2 x z^2 + 3 a^8 b^4 x z^2 - 8 a^6 b^6 x z^2 +
a^4 b^8 x z^2 + b^12 x z^2 + 28 a^10 c^2 x z^2 -
34 a^8 b^2 c^2 x z^2 + 4 a^4 b^6 c^2 x z^2 + 4 a^2 b^8 c^2 x z^2 -
2 b^10 c^2 x z^2 - 53 a^8 c^4 x z^2 + 48 a^6 b^2 c^4 x z^2 +
46 a^4 b^4 c^4 x z^2 - 24 a^2 b^6 c^4 x z^2 - 17 b^8 c^4 x z^2 +
40 a^6 c^6 x z^2 - 44 a^4 b^2 c^6 x z^2 + 36 b^6 c^6 x z^2 -
7 a^4 c^8 x z^2 + 24 a^2 b^2 c^8 x z^2 - 17 b^4 c^8 x z^2 -
4 a^2 c^10 x z^2 - 2 b^2 c^10 x z^2 + c^12 x z^2 + a^12 y z^2 +
a^8 b^4 y z^2 - 8 a^6 b^6 y z^2 + 3 a^4 b^8 y z^2 +
8 a^2 b^10 y z^2 - 5 b^12 y z^2 - 2 a^10 c^2 y z^2 +
4 a^8 b^2 c^2 y z^2 + 4 a^6 b^4 c^2 y z^2 - 34 a^2 b^8 c^2 y z^2 +
28 b^10 c^2 y z^2 - 17 a^8 c^4 y z^2 - 24 a^6 b^2 c^4 y z^2 +
46 a^4 b^4 c^4 y z^2 + 48 a^2 b^6 c^4 y z^2 - 53 b^8 c^4 y z^2 +
36 a^6 c^6 y z^2 - 44 a^2 b^4 c^6 y z^2 + 40 b^6 c^6 y z^2 -
17 a^4 c^8 y z^2 + 24 a^2 b^2 c^8 y z^2 - 7 b^4 c^8 y z^2 -
2 a^2 c^10 y z^2 - 4 b^2 c^10 y z^2 + c^12 y z^2 - a^12 z^3 -
2 a^10 b^2 z^3 + a^8 b^4 z^3 + 4 a^6 b^6 z^3 + a^4 b^8 z^3 -
2 a^2 b^10 z^3 - b^12 z^3 + 4 a^10 c^2 z^3 + 4 a^8 b^2 c^2 z^3 -
8 a^6 b^4 c^2 z^3 - 8 a^4 b^6 c^2 z^3 + 4 a^2 b^8 c^2 z^3 +
4 b^10 c^2 z^3 + 7 a^8 c^4 z^3 + 36 a^6 b^2 c^4 z^3 +
42 a^4 b^4 c^4 z^3 + 36 a^2 b^6 c^4 z^3 + 7 b^8 c^4 z^3 -
40 a^6 c^6 z^3 - 120 a^4 b^2 c^6 z^3 - 120 a^2 b^4 c^6 z^3 -
40 b^6 c^6 z^3 + 53 a^4 c^8 z^3 + 110 a^2 b^2 c^8 z^3 +
53 b^4 c^8 z^3 - 28 a^2 c^10 z^3 - 28 b^2 c^10 z^3 + 5 c^12 z^3

Francisco Javier García Capitán
17 March 2012

1 σχόλιο:

  1. Suppose, O is the circumcenter of ABC.A2B2C2 be the circumcevian triangle of G wrt ABC. Clearly, A3B3C3 be the circumcevian triangle of H wrt A2B2C2. A1,B1,C1 be the midpoints of HA3,HB3,HC3.
    Using this problem of yours(http://anthrakitis.blogspot.in/2012/02/pp.html) we can prove that ABC and A1B1C1 are perspective.

    ΑπάντησηΔιαγραφή

 

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