Reflections of the Circumcircle in the cevians

Let ABC be a triangle and P a point. Let (O1),(O2), (O3) be the reflections of the circumcircle (O) in BC, CA, AB, resp.

Let (O11), (O22), (O33) be the reflections of (O1), (O2), (O3) in AP, BP, CP, resp., L1,L2,L3 the radical axes of [(O22),(O33)], [(O33),(O11)], [(O11), (O22)], resp. and M1,M2,M3 the parallels to L1,L2,L3 through A,B,C resp.

Which is the locus of P such that M1,M2,M3 are concurrent?

APH, 5 December 2011

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Locus:

(-a - b + c) (a - b + c) (-a + b + c) (a + b + c) (x + y +
z) (-a^4 c^4 x^3 y^2 + a^2 b^2 c^4 x^3 y^2 + 2 a^2 c^6 x^3 y^2 +
b^2 c^6 x^3 y^2 - c^8 x^3 y^2 - a^2 b^2 c^4 x^2 y^3 +
b^4 c^4 x^2 y^3 - a^2 c^6 x^2 y^3 - 2 b^2 c^6 x^2 y^3 +
c^8 x^2 y^3 - a^6 b^2 x^3 y z + 3 a^4 b^4 x^3 y z -
3 a^2 b^6 x^3 y z + b^8 x^3 y z + a^6 c^2 x^3 y z -
b^6 c^2 x^3 y z - 3 a^4 c^4 x^3 y z + 3 a^2 c^6 x^3 y z +
b^2 c^6 x^3 y z - c^8 x^3 y z - a^8 x^2 y^2 z +
2 a^6 b^2 x^2 y^2 z - 2 a^2 b^6 x^2 y^2 z + b^8 x^2 y^2 z +
2 a^6 c^2 x^2 y^2 z - 2 b^6 c^2 x^2 y^2 z - a^4 c^4 x^2 y^2 z +
b^4 c^4 x^2 y^2 z - a^8 x y^3 z + 3 a^6 b^2 x y^3 z -
3 a^4 b^4 x y^3 z + a^2 b^6 x y^3 z + a^6 c^2 x y^3 z -
b^6 c^2 x y^3 z + 3 b^4 c^4 x y^3 z - a^2 c^6 x y^3 z -
3 b^2 c^6 x y^3 z + c^8 x y^3 z + a^4 b^4 x^3 z^2 -
2 a^2 b^6 x^3 z^2 + b^8 x^3 z^2 - a^2 b^4 c^2 x^3 z^2 -
b^6 c^2 x^3 z^2 + a^8 x^2 y z^2 - 2 a^6 b^2 x^2 y z^2 +
a^4 b^4 x^2 y z^2 - 2 a^6 c^2 x^2 y z^2 - b^4 c^4 x^2 y z^2 +
2 a^2 c^6 x^2 y z^2 + 2 b^2 c^6 x^2 y z^2 - c^8 x^2 y z^2 -
a^4 b^4 x y^2 z^2 + 2 a^2 b^6 x y^2 z^2 - b^8 x y^2 z^2 +
2 b^6 c^2 x y^2 z^2 + a^4 c^4 x y^2 z^2 - 2 a^2 c^6 x y^2 z^2 -
2 b^2 c^6 x y^2 z^2 + c^8 x y^2 z^2 - a^8 y^3 z^2 +
2 a^6 b^2 y^3 z^2 - a^4 b^4 y^3 z^2 + a^6 c^2 y^3 z^2 +
a^4 b^2 c^2 y^3 z^2 + a^2 b^6 x^2 z^3 - b^8 x^2 z^3 +
a^2 b^4 c^2 x^2 z^3 + 2 b^6 c^2 x^2 z^3 - b^4 c^4 x^2 z^3 +
a^8 x y z^3 - a^6 b^2 x y z^3 + a^2 b^6 x y z^3 - b^8 x y z^3 -
3 a^6 c^2 x y z^3 + 3 b^6 c^2 x y z^3 + 3 a^4 c^4 x y z^3 -
3 b^4 c^4 x y z^3 - a^2 c^6 x y z^3 + b^2 c^6 x y z^3 +
a^8 y^2 z^3 - a^6 b^2 y^2 z^3 - 2 a^6 c^2 y^2 z^3 -
a^4 b^2 c^2 y^2 z^3 + a^4 c^4 y^2 z^3) (-a^2 c^4 x^4 y^2 +
b^2 c^4 x^4 y^2 + c^6 x^4 y^2 - c^6 x^3 y^3 + a^2 c^4 x^2 y^4 -
b^2 c^4 x^2 y^4 + c^6 x^2 y^4 + a^6 x^4 y z - 2 a^4 b^2 x^4 y z +
a^2 b^4 x^4 y z - 2 a^4 c^2 x^4 y z + 2 b^4 c^2 x^4 y z +
a^2 c^4 x^4 y z + 2 b^2 c^4 x^4 y z + 2 a^6 x^3 y^2 z -
3 a^4 b^2 x^3 y^2 z + b^6 x^3 y^2 z - 4 a^4 c^2 x^3 y^2 z +
4 a^2 b^2 c^2 x^3 y^2 z - 6 b^2 c^4 x^3 y^2 z + 2 c^6 x^3 y^2 z +
a^6 x^2 y^3 z - 3 a^2 b^4 x^2 y^3 z + 2 b^6 x^2 y^3 z +
4 a^2 b^2 c^2 x^2 y^3 z - 4 b^4 c^2 x^2 y^3 z -
6 a^2 c^4 x^2 y^3 z + 2 c^6 x^2 y^3 z + a^4 b^2 x y^4 z -
2 a^2 b^4 x y^4 z + b^6 x y^4 z + 2 a^4 c^2 x y^4 z -
2 b^4 c^2 x y^4 z + 2 a^2 c^4 x y^4 z + b^2 c^4 x y^4 z -
a^2 b^4 x^4 z^2 + b^6 x^4 z^2 + b^4 c^2 x^4 z^2 +
2 a^6 x^3 y z^2 - 4 a^4 b^2 x^3 y z^2 + 2 b^6 x^3 y z^2 -
3 a^4 c^2 x^3 y z^2 + 4 a^2 b^2 c^2 x^3 y z^2 -
6 b^4 c^2 x^3 y z^2 + c^6 x^3 y z^2 + 3 a^6 x^2 y^2 z^2 -
3 a^4 b^2 x^2 y^2 z^2 - 3 a^2 b^4 x^2 y^2 z^2 +
3 b^6 x^2 y^2 z^2 - 3 a^4 c^2 x^2 y^2 z^2 -
6 a^2 b^2 c^2 x^2 y^2 z^2 - 3 b^4 c^2 x^2 y^2 z^2 -
3 a^2 c^4 x^2 y^2 z^2 - 3 b^2 c^4 x^2 y^2 z^2 +
3 c^6 x^2 y^2 z^2 + 2 a^6 x y^3 z^2 - 4 a^2 b^4 x y^3 z^2 +
2 b^6 x y^3 z^2 - 6 a^4 c^2 x y^3 z^2 + 4 a^2 b^2 c^2 x y^3 z^2 -
3 b^4 c^2 x y^3 z^2 + c^6 x y^3 z^2 + a^6 y^4 z^2 -
a^4 b^2 y^4 z^2 + a^4 c^2 y^4 z^2 - b^6 x^3 z^3 + a^6 x^2 y z^3 -
6 a^2 b^4 x^2 y z^3 + 2 b^6 x^2 y z^3 + 4 a^2 b^2 c^2 x^2 y z^3 -
3 a^2 c^4 x^2 y z^3 - 4 b^2 c^4 x^2 y z^3 + 2 c^6 x^2 y z^3 +
2 a^6 x y^2 z^3 - 6 a^4 b^2 x y^2 z^3 + b^6 x y^2 z^3 +
4 a^2 b^2 c^2 x y^2 z^3 - 4 a^2 c^4 x y^2 z^3 -
3 b^2 c^4 x y^2 z^3 + 2 c^6 x y^2 z^3 - a^6 y^3 z^3 +
a^2 b^4 x^2 z^4 + b^6 x^2 z^4 - b^4 c^2 x^2 z^4 +
2 a^4 b^2 x y z^4 + 2 a^2 b^4 x y z^4 + a^4 c^2 x y z^4 +
b^4 c^2 x y z^4 - 2 a^2 c^4 x y z^4 - 2 b^2 c^4 x y z^4 +
c^6 x y z^4 + a^6 y^2 z^4 + a^4 b^2 y^2 z^4 - a^4 c^2 y^2 z^4)

Francisco Javier García Capitán
5 December 2011

Reflections of the Circumcircle in the sidelines

Let ABC be a triangle, (O1),(O2), (O3) the reflections of (O) in BC,CA,AB, resp., (O12),(O13) the reflections of (O1) in AC, AB, resp., (O21),(O23) the reflections of (O2) in BC, BA, resp. and (O31),(O32) the reflections of (O3) in CB, CA, resp.

The radical axes of [(O12),(O13)],[(O21),(O23)], [(O31),(O32)] concur at the isogonal conjugate of the Nine Point Circle Center X(54).

APH, 5 December 2011

Generalization:

Let ABC be a triangle, P a point, P1,P2,P3 the reflections of P in BC,CA,AB, resp., P12,P13 the reflections of P1 in in AC, AB, resp., P21,P23 the reflections of P2 in BC, BA, resp. and P31,P32 the reflections of P3 in CB, CA, resp.

Which is the locus of P such that the perpendicular bisectors of P12P13,P21P2, P31P32 are concurrent?

APH, 5 December 2011

Reflections of the Circumcircle in the cevians of the Incenter (2)

Let ABC be a triangle. Let (O1),(O2), (O3) be the reflections of the circumcircle (O) in the cevians AI, BI, CI, resp.

Let (O11), (O22), (O33) be the reflections of (O1), (O2), (O3) in BC, CA, AB, resp., L1,L2,L3 the radical axes of [(O22),(O33)], [(O33),(O11)], [(O11), (O22)], resp. and M1,M2,M3 the parallels to L1,L2,L3 through A,B,C resp.

The lines M1,M2,M3 are concurrent.

Point of concurrence?

Locus: For P instead of I, such that M1,M2,M3 are concurrent?

APH, 4 December 2011

The locus is McCay cubic, circumcircle, line at infinity and a sextic with no real points.

Francisco Javier García Capitán
5 December 2011

Nice, if true!

1. Let ABC be a triangle.

Denote:

(O1), (O2), (O3) the reflections of the circumcircle (O) in AI, BI, CI, rersp.

L1, L2, L3 the radical axes of [(O2), (O3)], [(O3), (O1)], [(O1), (O2)], resp.

M1, M2, M3 the reflections of L1, L2, L3 in BC, CA, AB, resp.

Are the lines M1 ,M2, M3 concurrent? Point?


2. Let ABC be a triangle.

Denote:

(O1), (O2), (O3) the reflections of the circumcircle (O) in AI, BI, CI, rersp.

(O12), (O13) the reflections of (O1) in BI, CI, resp.

(O23), (O21) the reflections of (O2) in CI, AI, resp.

(O31), (O32) the reflections of (O3) in AI, BI, resp.

L1, L2, L3 the radical axes of [(O12), (O13)], [(O23), (O21)], [(O31), (O32)], resp.

M1, M2, M3 the reflections of L1, L2, L3 in BC, CA, AB, resp.

Are the lines M1 ,M2, M3 concurrent? Point?

Addendum: Loci ?? (P instead of I)

APH, 3 December 2011

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1. P=(x:y:z) can be any point.

The point of concurrence is:
{a^2 (b^2 c^2 x^2 + a^2 c^2 x y - c^4 x y + a^2 b^2 x z - b^4 x z +
a^4 y z - a^2 b^2 y z - a^2 c^2 y z),
b^2 (b^2 c^2 x y - c^4 x y + a^2 c^2 y^2 - a^2 b^2 x z + b^4 x z -
b^2 c^2 x z - a^4 y z + a^2 b^2 y z),
c^2 (-a^2 c^2 x y - b^2 c^2 x y + c^4 x y - b^4 x z + b^2 c^2 x z -
a^4 y z + a^2 c^2 y z + a^2 b^2 z^2)}

2. The locus of P is circumcircle, line at infinity and the curve:

-a^2 b^2 c^4 x^3 y - b^4 c^4 x^3 y + b^2 c^6 x^3 y + a^4 c^4 x y^3 +
a^2 b^2 c^4 x y^3 - a^2 c^6 x y^3 + a^2 b^4 c^2 x^3 z -
b^6 c^2 x^3 z + b^4 c^4 x^3 z + a^6 c^2 y^3 z - a^4 b^2 c^2 y^3 z -
a^4 c^4 y^3 z - a^4 b^4 x z^3 + a^2 b^6 x z^3 - a^2 b^4 c^2 x z^3 -
a^6 b^2 y z^3 + a^4 b^4 y z^3 + a^4 b^2 c^2 y z^3

through X1, X3, X6

Francisco Javier García Capitán
3 December 2011

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This is the Euler-Morley quartic Q002.

Bernard Gibert
4 December 2011

Reflections of the Circumcircle in the cevians of the Incenter

Let ABC be a triangle. Let (O1),(O2), (O3) be the reflections of the circumcircle (O) in the cevians AI, BI, CI, resp

Let L1 be the common chord (radical axis) of (O2) and (O3) and similarly L2 and L3 (concurrent at I). Let M1, M2, M3 be the parallels to L1,L2,L3 through A,B,C, resp. The lines M1,M2,M3 are concurent.

Point of concurrence?

Generalization:
P instead of I. Locus of P such that M1,M2,M3 are concurrent?

APH, 2 December 2011

The locus of P is: Line at Infinity + Circumcircle + McCay cubic.

Francisco Javier García Capitán
2 December 2011

Reflections of cevians in cevians

Lert ABC be a triangle.

A* := (Reflection of AH in BI) /\ (Reflection of AH in CI)

B* := (Reflection of BH in CI) /\ (Reflection of BH in AI)

C* := (Reflection of CH in AI) /\ (Reflection of CH in BI)

The triangles ABC, A*B*C* are perspective.

Perspector?

Generalization:

P, Q instead of H,I
Locus of P such that ABC, A*B*C* are perspective?

APH, 1 December 2011

For H, I, the perspector is X84.

In the general case, the equaition of the locus for P=(x,y,z)and Q=(u,v,w) is long (see below).

Francisco Javier García Capitán
2 December 2011

Reflections of a line through H in parallels to altitudes through A,B,C

Let ABC be a triangle, HaHbHc the orthic triangle, L a line through H and Lab,Lac the parallels through A to altitudes BHb, CHc, resp. The reflections of L in Lab, Lac intersect at A'. Similarly B',C'.

The triangles ABC, A'B'C' are perspective. The perspector lies on the circumcircle of ABC, where concur the reflections of L in the sidelines of ABC.

Note: The triangle A'B'C' is also perspective with the antimedial and medial triangles of ABC.

APH, 25 November 2011