#4584
Dear geometers,
Let ABC be a triangle.
O and H are circumcenter and orthocenter.
I is incenter of ABC.
Ja, Jb, Jc are incenters of triangles AHO, BHO, CHO.
Then isogonal conjugate of I wrt triangle JaJbJc lies on line OH.
Which is this point?
Best regards,
Tran Quang Hung.
---------------------------------------------
#4585
#4585
I find huge coordinates ( T stands for S·OH where S is twice the area of ABC.
Best regards,
Francisco Javier.
#4587
{-a^13 + a^12 b + 3 a^11 b^2 - 3 a^10 b^3 - 2 a^9 b^4 + 2 a^8 b^5 -
2 a^7 b^6 + 2 a^6 b^7 + 3 a^5 b^8 - 3 a^4 b^9 - a^3 b^10 +
a^2 b^11 + a^12 c - 2 a^11 b c - 3 a^10 b^2 c + 2 a^9 b^3 c +
3 a^8 b^4 c + 4 a^7 b^5 c - a^6 b^6 c - 4 a^5 b^7 c - 2 a^3 b^9 c +
2 a b^11 c + 3 a^11 c^2 - 3 a^10 b c^2 - 5 a^9 b^2 c^2 +
4 a^8 b^3 c^2 + 4 a^7 b^4 c^2 - 2 a^6 b^5 c^2 - 4 a^5 b^6 c^2 +
4 a^4 b^7 c^2 + a^3 b^8 c^2 - 3 a^2 b^9 c^2 + a b^10 c^2 -
3 a^10 c^3 + 2 a^9 b c^3 + 4 a^8 b^2 c^3 - 10 a^7 b^3 c^3 -
3 a^6 b^4 c^3 + 4 a^5 b^5 c^3 + 2 a^4 b^6 c^3 + 10 a^3 b^7 c^3 -
6 a b^9 c^3 - 2 a^9 c^4 + 3 a^8 b c^4 + 4 a^7 b^2 c^4 -
3 a^6 b^3 c^4 + 2 a^5 b^4 c^4 - 3 a^4 b^5 c^4 + 3 a^2 b^7 c^4 -
4 a b^8 c^4 + 2 a^8 c^5 + 4 a^7 b c^5 - 2 a^6 b^2 c^5 +
4 a^5 b^3 c^5 - 3 a^4 b^4 c^5 - 16 a^3 b^5 c^5 - a^2 b^6 c^5 +
4 a b^7 c^5 - 2 a^7 c^6 - a^6 b c^6 - 4 a^5 b^2 c^6 +
2 a^4 b^3 c^6 - a^2 b^5 c^6 + 6 a b^6 c^6 + 2 a^6 c^7 -
4 a^5 b c^7 + 4 a^4 b^2 c^7 + 10 a^3 b^3 c^7 + 3 a^2 b^4 c^7 +
4 a b^5 c^7 + 3 a^5 c^8 + a^3 b^2 c^8 - 4 a b^4 c^8 - 3 a^4 c^9 -
2 a^3 b c^9 - 3 a^2 b^2 c^9 - 6 a b^3 c^9 - a^3 c^10 + a b^2 c^10 +
a^2 c^11 + 2 a b c^11 + 2 a^10 T - 4 a^9 b T - 4 a^8 b^2 T +
10 a^7 b^3 T - 6 a^5 b^5 T + 4 a^4 b^6 T - 2 a^3 b^7 T -
2 a^2 b^8 T + 2 a b^9 T - 4 a^9 c T + 6 a^8 b c T - 4 a^6 b^3 c T +
8 a^5 b^4 c T - 8 a^4 b^5 c T + 4 a^2 b^7 c T - 4 a b^8 c T +
2 b^9 c T - 4 a^8 c^2 T + 4 a^6 b^2 c^2 T - 10 a^5 b^3 c^2 T -
4 a^4 b^4 c^2 T + 12 a^3 b^5 c^2 T + 4 a^2 b^6 c^2 T -
2 a b^7 c^2 T + 10 a^7 c^3 T - 4 a^6 b c^3 T - 10 a^5 b^2 c^3 T +
16 a^4 b^3 c^3 T - 10 a^3 b^4 c^3 T - 4 a^2 b^5 c^3 T +
10 a b^6 c^3 T - 8 b^7 c^3 T + 8 a^5 b c^4 T - 4 a^4 b^2 c^4 T -
10 a^3 b^3 c^4 T - 4 a^2 b^4 c^4 T - 6 a b^5 c^4 T - 6 a^5 c^5 T -
8 a^4 b c^5 T + 12 a^3 b^2 c^5 T - 4 a^2 b^3 c^5 T -
6 a b^4 c^5 T + 12 b^5 c^5 T + 4 a^4 c^6 T + 4 a^2 b^2 c^6 T +
10 a b^3 c^6 T - 2 a^3 c^7 T + 4 a^2 b c^7 T - 2 a b^2 c^7 T -
8 b^3 c^7 T - 2 a^2 c^8 T - 4 a b c^8 T + 2 a c^9 T + 2 b c^9 T,
a^11 b^2 - a^10 b^3 - 3 a^9 b^4 + 3 a^8 b^5 + 2 a^7 b^6 -
2 a^6 b^7 + 2 a^5 b^8 - 2 a^4 b^9 - 3 a^3 b^10 + 3 a^2 b^11 +
a b^12 - b^13 + 2 a^11 b c - 2 a^9 b^3 c - 4 a^7 b^5 c -
a^6 b^6 c + 4 a^5 b^7 c + 3 a^4 b^8 c + 2 a^3 b^9 c -
3 a^2 b^10 c - 2 a b^11 c + b^12 c + a^10 b c^2 - 3 a^9 b^2 c^2 +
a^8 b^3 c^2 + 4 a^7 b^4 c^2 - 4 a^6 b^5 c^2 - 2 a^5 b^6 c^2 +
4 a^4 b^7 c^2 + 4 a^3 b^8 c^2 - 5 a^2 b^9 c^2 - 3 a b^10 c^2 +
3 b^11 c^2 - 6 a^9 b c^3 + 10 a^7 b^3 c^3 + 2 a^6 b^4 c^3 +
4 a^5 b^5 c^3 - 3 a^4 b^6 c^3 - 10 a^3 b^7 c^3 + 4 a^2 b^8 c^3 +
2 a b^9 c^3 - 3 b^10 c^3 - 4 a^8 b c^4 + 3 a^7 b^2 c^4 -
3 a^5 b^4 c^4 + 2 a^4 b^5 c^4 - 3 a^3 b^6 c^4 + 4 a^2 b^7 c^4 +
3 a b^8 c^4 - 2 b^9 c^4 + 4 a^7 b c^5 - a^6 b^2 c^5 -
16 a^5 b^3 c^5 - 3 a^4 b^4 c^5 + 4 a^3 b^5 c^5 - 2 a^2 b^6 c^5 +
4 a b^7 c^5 + 2 b^8 c^5 + 6 a^6 b c^6 - a^5 b^2 c^6 +
2 a^3 b^4 c^6 - 4 a^2 b^5 c^6 - a b^6 c^6 - 2 b^7 c^6 +
4 a^5 b c^7 + 3 a^4 b^2 c^7 + 10 a^3 b^3 c^7 + 4 a^2 b^4 c^7 -
4 a b^5 c^7 + 2 b^6 c^7 - 4 a^4 b c^8 + a^2 b^3 c^8 + 3 b^5 c^8 -
6 a^3 b c^9 - 3 a^2 b^2 c^9 - 2 a b^3 c^9 - 3 b^4 c^9 + a^2 b c^10 -
b^3 c^10 + 2 a b c^11 + b^2 c^11 + 2 a^9 b T - 2 a^8 b^2 T -
2 a^7 b^3 T + 4 a^6 b^4 T - 6 a^5 b^5 T + 10 a^3 b^7 T -
4 a^2 b^8 T - 4 a b^9 T + 2 b^10 T + 2 a^9 c T - 4 a^8 b c T +
4 a^7 b^2 c T - 8 a^5 b^4 c T + 8 a^4 b^5 c T - 4 a^3 b^6 c T +
6 a b^8 c T - 4 b^9 c T - 2 a^7 b c^2 T + 4 a^6 b^2 c^2 T +
12 a^5 b^3 c^2 T - 4 a^4 b^4 c^2 T - 10 a^3 b^5 c^2 T +
4 a^2 b^6 c^2 T - 4 b^8 c^2 T - 8 a^7 c^3 T + 10 a^6 b c^3 T -
4 a^5 b^2 c^3 T - 10 a^4 b^3 c^3 T + 16 a^3 b^4 c^3 T -
10 a^2 b^5 c^3 T - 4 a b^6 c^3 T + 10 b^7 c^3 T - 6 a^5 b c^4 T -
4 a^4 b^2 c^4 T - 10 a^3 b^3 c^4 T - 4 a^2 b^4 c^4 T +
8 a b^5 c^4 T + 12 a^5 c^5 T - 6 a^4 b c^5 T - 4 a^3 b^2 c^5 T +
12 a^2 b^3 c^5 T - 8 a b^4 c^5 T - 6 b^5 c^5 T + 10 a^3 b c^6 T +
4 a^2 b^2 c^6 T + 4 b^4 c^6 T - 8 a^3 c^7 T - 2 a^2 b c^7 T +
4 a b^2 c^7 T - 2 b^3 c^7 T - 4 a b c^8 T - 2 b^2 c^8 T +
2 a c^9 T + 2 b c^9 T,
2 a^11 b c + a^10 b^2 c - 6 a^9 b^3 c - 4 a^8 b^4 c + 4 a^7 b^5 c +
6 a^6 b^6 c + 4 a^5 b^7 c - 4 a^4 b^8 c - 6 a^3 b^9 c +
a^2 b^10 c + 2 a b^11 c + a^11 c^2 - 3 a^9 b^2 c^2 + 3 a^7 b^4 c^2 -
a^6 b^5 c^2 - a^5 b^6 c^2 + 3 a^4 b^7 c^2 - 3 a^2 b^9 c^2 +
b^11 c^2 - a^10 c^3 - 2 a^9 b c^3 + a^8 b^2 c^3 + 10 a^7 b^3 c^3 -
16 a^5 b^5 c^3 + 10 a^3 b^7 c^3 + a^2 b^8 c^3 - 2 a b^9 c^3 -
b^10 c^3 - 3 a^9 c^4 + 4 a^7 b^2 c^4 + 2 a^6 b^3 c^4 -
3 a^5 b^4 c^4 - 3 a^4 b^5 c^4 + 2 a^3 b^6 c^4 + 4 a^2 b^7 c^4 -
3 b^9 c^4 + 3 a^8 c^5 - 4 a^7 b c^5 - 4 a^6 b^2 c^5 +
4 a^5 b^3 c^5 + 2 a^4 b^4 c^5 + 4 a^3 b^5 c^5 - 4 a^2 b^6 c^5 -
4 a b^7 c^5 + 3 b^8 c^5 + 2 a^7 c^6 - a^6 b c^6 - 2 a^5 b^2 c^6 -
3 a^4 b^3 c^6 - 3 a^3 b^4 c^6 - 2 a^2 b^5 c^6 - a b^6 c^6 +
2 b^7 c^6 - 2 a^6 c^7 + 4 a^5 b c^7 + 4 a^4 b^2 c^7 -
10 a^3 b^3 c^7 + 4 a^2 b^4 c^7 + 4 a b^5 c^7 - 2 b^6 c^7 +
2 a^5 c^8 + 3 a^4 b c^8 + 4 a^3 b^2 c^8 + 4 a^2 b^3 c^8 +
3 a b^4 c^8 + 2 b^5 c^8 - 2 a^4 c^9 + 2 a^3 b c^9 - 5 a^2 b^2 c^9 +
2 a b^3 c^9 - 2 b^4 c^9 - 3 a^3 c^10 - 3 a^2 b c^10 -
3 a b^2 c^10 - 3 b^3 c^10 + 3 a^2 c^11 - 2 a b c^11 + 3 b^2 c^11 +
a c^12 + b c^12 - c^13 + 2 a^9 b T - 8 a^7 b^3 T + 12 a^5 b^5 T -
8 a^3 b^7 T + 2 a b^9 T + 2 a^9 c T - 4 a^8 b c T - 2 a^7 b^2 c T +
10 a^6 b^3 c T - 6 a^5 b^4 c T - 6 a^4 b^5 c T + 10 a^3 b^6 c T -
2 a^2 b^7 c T - 4 a b^8 c T + 2 b^9 c T - 2 a^8 c^2 T +
4 a^7 b c^2 T + 4 a^6 b^2 c^2 T - 4 a^5 b^3 c^2 T -
4 a^4 b^4 c^2 T - 4 a^3 b^5 c^2 T + 4 a^2 b^6 c^2 T +
4 a b^7 c^2 T - 2 b^8 c^2 T - 2 a^7 c^3 T + 12 a^5 b^2 c^3 T -
10 a^4 b^3 c^3 T - 10 a^3 b^4 c^3 T + 12 a^2 b^5 c^3 T -
2 b^7 c^3 T + 4 a^6 c^4 T - 8 a^5 b c^4 T - 4 a^4 b^2 c^4 T +
16 a^3 b^3 c^4 T - 4 a^2 b^4 c^4 T - 8 a b^5 c^4 T + 4 b^6 c^4 T -
6 a^5 c^5 T + 8 a^4 b c^5 T - 10 a^3 b^2 c^5 T - 10 a^2 b^3 c^5 T +
8 a b^4 c^5 T - 6 b^5 c^5 T - 4 a^3 b c^6 T + 4 a^2 b^2 c^6 T -
4 a b^3 c^6 T + 10 a^3 c^7 T + 10 b^3 c^7 T - 4 a^2 c^8 T +
6 a b c^8 T - 4 b^2 c^8 T - 4 a c^9 T - 4 b c^9 T + 2 c^10 T}
Frabcisco Javier Garcia Capitan
---------------------------------------------
#4587
Dear Mr Francisco and friends,
I see similar problem with excenters.
H,O and I are orthocenter, circumcenter and incenter of ABC.
If Ja, Jb, Jc are incenters of AOH, BOH and COH.
If Ja’, Jb’, Jc’ are A,B,C excenters of AOH, BOH, COH.
I1 is isogonal conjugate of I wrt triangle JaJbJc.
I2 is isogonal conjugate of I wrt triangle Ja’Jb’Jc’.
Then I1, I2 are on line OH and they are reflection in NPC center of ABC.
Best regards,
Tran Quang Hung.
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου