Πέμπτη, 26 Ιανουαρίου 2012

LOCUS


Let ABC be triangle, P a point, A1B1C1 the cevian triangle of P and A2B2C2 the circumcevian triangle of P.


Denote:

Ab = the other than A2 intersection of the circumcircle and A2B1
Ac = the other than A2 intersection of the circumcircle and A2C1

Bc = the other than B2 intersection of the circumcircle and B2C1
Ba = the other than B2 intersection of the circumcircle and B2A1

Ca = the other than C2 intersection of the circumcircle and C2A1
Cb = the other than C2 intersection of the circumcircle and C2B1

The lines AbAc, BcBa, CaCb bound a triangle A3B3C3.

Which is the locus of P such that:

1. ABC, A3B3C3

2. A1B1C1, A3B3C3

3. A2B2C2, A3B3C3

are perspective?


APH, 26 January 2012

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For 1. and 3. the locus is the whole plane.
For 2, the locus is (symmedians) + circumcircle + (12th degree curve whose isogonal conjugate is 6th degree curve).

For 1. the perspector is

{a^2 x (c^2 y + b^2 z) (c^4 x^2 y^2 + b^2 c^2 x^2 y z +
2 a^2 c^2 x y^2 z + b^4 x^2 z^2 + a^2 b^2 x y z^2 +
a^4 y^2 z^2) (c^4 x^2 y^2 + b^2 c^2 x^2 y z + a^2 c^2 x y^2 z +
b^4 x^2 z^2 + 2 a^2 b^2 x y z^2 + a^4 y^2 z^2),
b^2 y (c^2 x + a^2 z) (c^4 x^2 y^2 + 2 b^2 c^2 x^2 y z +
a^2 c^2 x y^2 z + b^4 x^2 z^2 + a^2 b^2 x y z^2 +
a^4 y^2 z^2) (c^4 x^2 y^2 + b^2 c^2 x^2 y z + a^2 c^2 x y^2 z +
b^4 x^2 z^2 + 2 a^2 b^2 x y z^2 + a^4 y^2 z^2),
c^2 (b^2 x + a^2 y) z (c^4 x^2 y^2 + 2 b^2 c^2 x^2 y z +
a^2 c^2 x y^2 z + b^4 x^2 z^2 + a^2 b^2 x y z^2 +
a^4 y^2 z^2) (c^4 x^2 y^2 + b^2 c^2 x^2 y z + 2 a^2 c^2 x y^2 z +
b^4 x^2 z^2 + a^2 b^2 x y z^2 + a^4 y^2 z^2)}

For 2. the perspector is

{a^2 (-c^16 x^8 y^8 - 6 c^14 x^7 y^7 (b^2 x + a^2 y) z -
c^12 x^6 y^6 (17 b^4 x^2 + 30 a^2 b^2 x y + 14 a^4 y^2) z^2 -
c^10 x^5 y^5 (30 b^6 x^3 + 69 a^2 b^4 x^2 y + 54 a^4 b^2 x y^2 +
14 a^6 y^3) z^3 -
3 b^2 c^8 x^5 y^4 (12 b^6 x^3 + 33 a^2 b^4 x^2 y +
31 a^4 b^2 x y^2 + 10 a^6 y^3) z^4 -
c^6 x^3 y^3 (30 b^10 x^5 + 99 a^2 b^8 x^4 y +
107 a^4 b^6 x^3 y^2 + 21 a^6 b^4 x^2 y^3 - 30 a^8 b^2 x y^4 -
14 a^10 y^5) z^5 -
c^4 x^2 y^2 (b^2 x + a^2 y)^3 (17 b^6 x^3 + 18 a^2 b^4 x^2 y -
12 a^4 b^2 x y^2 - 14 a^6 y^3) z^6 -
6 c^2 x y (b^2 x - a^2 y) (b^2 x + a^2 y)^6 z^7 - (b^2 x -
a^2 y) (b^2 x + a^2 y)^7 z^8),
b^2 (-c^16 x^8 y^8 - 6 c^14 x^7 y^7 (b^2 x + a^2 y) z -
c^12 x^6 y^6 (14 b^4 x^2 + 30 a^2 b^2 x y + 17 a^4 y^2) z^2 -
c^10 x^5 y^5 (14 b^6 x^3 + 54 a^2 b^4 x^2 y + 69 a^4 b^2 x y^2 +
30 a^6 y^3) z^3 -
3 a^2 c^8 x^4 y^5 (10 b^6 x^3 + 31 a^2 b^4 x^2 y +
33 a^4 b^2 x y^2 + 12 a^6 y^3) z^4 +
c^6 x^3 y^3 (14 b^10 x^5 + 30 a^2 b^8 x^4 y - 21 a^4 b^6 x^3 y^2 -
107 a^6 b^4 x^2 y^3 - 99 a^8 b^2 x y^4 - 30 a^10 y^5) z^5 +
c^4 x^2 y^2 (b^2 x + a^2 y)^3 (14 b^6 x^3 + 12 a^2 b^4 x^2 y -
18 a^4 b^2 x y^2 - 17 a^6 y^3) z^6 +
6 c^2 x y (b^2 x - a^2 y) (b^2 x + a^2 y)^6 z^7 + (b^2 x -
a^2 y) (b^2 x + a^2 y)^7 z^8),
c^2 (c^16 x^8 y^8 + 6 c^14 x^7 y^7 (b^2 x + a^2 y) z +
2 c^12 x^6 y^6 (7 b^4 x^2 + 15 a^2 b^2 x y + 7 a^4 y^2) z^2 +
2 c^10 x^5 y^5 (7 b^6 x^3 + 27 a^2 b^4 x^2 y + 27 a^4 b^2 x y^2 +
7 a^6 y^3) z^3 + 30 c^8 x^5 y^5 (a b^3 x + a^3 b y)^2 z^4 -
c^6 x^3 y^3 (14 b^10 x^5 + 30 a^2 b^8 x^4 y +
21 a^4 b^6 x^3 y^2 + 21 a^6 b^4 x^2 y^3 + 30 a^8 b^2 x y^4 +
14 a^10 y^5) z^5 -
c^4 x^2 y^2 (14 b^12 x^6 + 54 a^2 b^10 x^5 y +
93 a^4 b^8 x^4 y^2 + 107 a^6 b^6 x^3 y^3 +
93 a^8 b^4 x^2 y^4 + 54 a^10 b^2 x y^5 + 14 a^12 y^6) z^6 -
3 c^2 x y (b^2 x + a^2 y)^3 (2 b^8 x^4 + 4 a^2 b^6 x^3 y +
5 a^4 b^4 x^2 y^2 + 4 a^6 b^2 x y^3 + 2 a^8 y^4) z^7 - (b^2 x +
a^2 y)^4 (b^4 x^2 + a^2 b^2 x y + a^4 y^2)^2 z^8)}

For 3. the equation of the 12th curve is

c^12 x^6 y^6 + 5 b^2 c^10 x^6 y^5 z + 5 a^2 c^10 x^5 y^6 z +
11 b^4 c^8 x^6 y^4 z^2 + 23 a^2 b^2 c^8 x^5 y^5 z^2 +
11 a^4 c^8 x^4 y^6 z^2 + 14 b^6 c^6 x^6 y^3 z^3 +
44 a^2 b^4 c^6 x^5 y^4 z^3 + 44 a^4 b^2 c^6 x^4 y^5 z^3 +
14 a^6 c^6 x^3 y^6 z^3 + 11 b^8 c^4 x^6 y^2 z^4 +
44 a^2 b^6 c^4 x^5 y^3 z^4 + 65 a^4 b^4 c^4 x^4 y^4 z^4 +
44 a^6 b^2 c^4 x^3 y^5 z^4 + 11 a^8 c^4 x^2 y^6 z^4 +
5 b^10 c^2 x^6 y z^5 + 23 a^2 b^8 c^2 x^5 y^2 z^5 +
44 a^4 b^6 c^2 x^4 y^3 z^5 + 44 a^6 b^4 c^2 x^3 y^4 z^5 +
23 a^8 b^2 c^2 x^2 y^5 z^5 + 5 a^10 c^2 x y^6 z^5 + b^12 x^6 z^6 +
5 a^2 b^10 x^5 y z^6 + 11 a^4 b^8 x^4 y^2 z^6 +
14 a^6 b^6 x^3 y^3 z^6 + 11 a^8 b^4 x^2 y^4 z^6 +
5 a^10 b^2 x y^5 z^6 + a^12 y^6 z^6 = 0.

Francisco Javier García Capitán
27 January 2012

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Variation:

Let ABC be triangle, P a point, A1B1C1 the pedal (instead of cevian) triangle of P and A2B2C2 the circumcevian triangle of P. etc

APH


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