## Δευτέρα, 12 Δεκεμβρίου 2011

### Parallel Lines : GENERALIZATION 2

[APH]:
> > Let ABC be a triangle and L a line.

> >
> > L1,L2,L3 := the reflections of AI, BI, CI in L, resp.
> >
> > M1,M2,M3 := the reflections of AH, BH, CH in L1, L2, L3, resp
then M1,M2,M3 are parallel.

[Jean-Pierre Ehrmann]:
If (L,L') is the directed angle (mod Pi) between the lines L & L', we have
(M1,M2)+(HA,HB)=2(L1,L2)=2(BI,AI)=(CB,CA)=(HA,HB) thus (M1,M2)=0

Something curious : it seems that, if we take X(80) (reflection of the incenter in the Feuerbach point) instead of H, the lines M1,M2,M3 concur for every line L. But why?

Hyacinthos #20524

[Francisco Javier]:

[APH]:
>It seems that H and X(80) are points of the locus:
>Let ABC be a triangle P a variable point and L a fixed line.
>Let L1, L2, L3 be the reflections of AI,BI,CI in L and
>M1, M2, M3 the reflections of AP,BP,CP in L1,L2,L3, resp.
>Which is the locus of P such that M1,M2,M3 are concurrent?

For each line L: u x + v y + w z = 0 the locus of P=(x:y:z) is a cubic (see
below).

Both X4 and X80 lie on any of these cubics. From a quick sketch, these are the only points lying on all cubics.

Equation of cubic for L: u x + v y + w z = 0 is as follows:

a^5 u^2 x^2 y - 2 a^3 b^2 u^2 x^2 y + a b^4 u^2 x^2 y +
a^3 b c u^2 x^2 y - a b^3 c u^2 x^2 y + a b c^3 u^2 x^2 y -
a c^4 u^2 x^2 y - a^5 u v x^2 y + a^4 b u v x^2 y +
2 a^3 b^2 u v x^2 y - 2 a^2 b^3 u v x^2 y - a b^4 u v x^2 y +
b^5 u v x^2 y - 2 a^3 b c u v x^2 y + 2 a b^3 c u v x^2 y +
2 a^3 c^2 u v x^2 y + a^2 b c^2 u v x^2 y - 2 a b^2 c^2 u v x^2 y -
b^3 c^2 u v x^2 y - a c^4 u v x^2 y - a^4 b v^2 x^2 y +
2 a^2 b^3 v^2 x^2 y - b^5 v^2 x^2 y + a^3 b c v^2 x^2 y -
a b^3 c v^2 x^2 y + a^2 b c^2 v^2 x^2 y - b^3 c^2 v^2 x^2 y -
a b c^3 v^2 x^2 y - 2 a^3 c^2 u w x^2 y + 2 a b^2 c^2 u w x^2 y +
a^2 c^3 u w x^2 y - 2 a b c^3 u w x^2 y + b^2 c^3 u w x^2 y +
2 a c^4 u w x^2 y - c^5 u w x^2 y - 2 a^2 b c^2 v w x^2 y +
2 b^3 c^2 v w x^2 y + 2 a b c^3 v w x^2 y - 2 b^2 c^3 v w x^2 y -
a^2 c^3 w^2 x^2 y + b^2 c^3 w^2 x^2 y + c^5 w^2 x^2 y +
a^5 u^2 x y^2 - 2 a^3 b^2 u^2 x y^2 + a b^4 u^2 x y^2 +
a^3 b c u^2 x y^2 - a b^3 c u^2 x y^2 + a^3 c^2 u^2 x y^2 -
a b^2 c^2 u^2 x y^2 + a b c^3 u^2 x y^2 - a^5 u v x y^2 +
a^4 b u v x y^2 + 2 a^3 b^2 u v x y^2 - 2 a^2 b^3 u v x y^2 -
a b^4 u v x y^2 + b^5 u v x y^2 - 2 a^3 b c u v x y^2 +
2 a b^3 c u v x y^2 + a^3 c^2 u v x y^2 + 2 a^2 b c^2 u v x y^2 -
a b^2 c^2 u v x y^2 - 2 b^3 c^2 u v x y^2 + b c^4 u v x y^2 -
a^4 b v^2 x y^2 + 2 a^2 b^3 v^2 x y^2 - b^5 v^2 x y^2 +
a^3 b c v^2 x y^2 - a b^3 c v^2 x y^2 - a b c^3 v^2 x y^2 +
b c^4 v^2 x y^2 - 2 a^3 c^2 u w x y^2 + 2 a b^2 c^2 u w x y^2 +
2 a^2 c^3 u w x y^2 - 2 a b c^3 u w x y^2 - 2 a^2 b c^2 v w x y^2 +
2 b^3 c^2 v w x y^2 - a^2 c^3 v w x y^2 + 2 a b c^3 v w x y^2 -
b^2 c^3 v w x y^2 - 2 b c^4 v w x y^2 + c^5 v w x y^2 -
a^2 c^3 w^2 x y^2 + b^2 c^3 w^2 x y^2 - c^5 w^2 x y^2 -
a^5 u^2 x^2 z + a b^4 u^2 x^2 z - a^3 b c u^2 x^2 z -
a b^3 c u^2 x^2 z + 2 a^3 c^2 u^2 x^2 z + a b c^3 u^2 x^2 z -
a c^4 u^2 x^2 z + 2 a^3 b^2 u v x^2 z - a^2 b^3 u v x^2 z -
2 a b^4 u v x^2 z + b^5 u v x^2 z + 2 a b^3 c u v x^2 z -
2 a b^2 c^2 u v x^2 z - b^3 c^2 u v x^2 z + a^2 b^3 v^2 x^2 z -
b^5 v^2 x^2 z - b^3 c^2 v^2 x^2 z + a^5 u w x^2 z -
2 a^3 b^2 u w x^2 z + a b^4 u w x^2 z - a^4 c u w x^2 z +
2 a^3 b c u w x^2 z - a^2 b^2 c u w x^2 z - 2 a^3 c^2 u w x^2 z +
2 a b^2 c^2 u w x^2 z + 2 a^2 c^3 u w x^2 z - 2 a b c^3 u w x^2 z +
b^2 c^3 u w x^2 z + a c^4 u w x^2 z - c^5 u w x^2 z +
2 a^2 b^2 c v w x^2 z - 2 a b^3 c v w x^2 z + 2 b^3 c^2 v w x^2 z -
2 b^2 c^3 v w x^2 z + a^4 c w^2 x^2 z - a^3 b c w^2 x^2 z -
a^2 b^2 c w^2 x^2 z + a b^3 c w^2 x^2 z - 2 a^2 c^3 w^2 x^2 z +
a b c^3 w^2 x^2 z + b^2 c^3 w^2 x^2 z + c^5 w^2 x^2 z -
a^3 b^2 u^2 x y z + a b^4 u^2 x y z - 2 a b^3 c u^2 x y z +
a^3 c^2 u^2 x y z + 2 a b c^3 u^2 x y z - a c^4 u^2 x y z -
a^5 u v x y z + 2 a^4 b u v x y z + 3 a^3 b^2 u v x y z -
3 a^2 b^3 u v x y z - 2 a b^4 u v x y z + b^5 u v x y z -
4 a^3 b c u v x y z + 4 a b^3 c u v x y z + 2 a^3 c^2 u v x y z +
a^2 b c^2 u v x y z - a b^2 c^2 u v x y z - 2 b^3 c^2 u v x y z -
a c^4 u v x y z + b c^4 u v x y z - a^4 b v^2 x y z +
a^2 b^3 v^2 x y z + 2 a^3 b c v^2 x y z - b^3 c^2 v^2 x y z -
2 a b c^3 v^2 x y z + b c^4 v^2 x y z + a^5 u w x y z -
2 a^3 b^2 u w x y z + a b^4 u w x y z - 2 a^4 c u w x y z +
4 a^3 b c u w x y z - a^2 b^2 c u w x y z - b^4 c u w x y z -
3 a^3 c^2 u w x y z + a b^2 c^2 u w x y z + 3 a^2 c^3 u w x y z -
4 a b c^3 u w x y z + 2 b^2 c^3 u w x y z + 2 a c^4 u w x y z -
c^5 u w x y z - a^4 b v w x y z + 2 a^2 b^3 v w x y z -
b^5 v w x y z + a^4 c v w x y z + a^2 b^2 c v w x y z -
4 a b^3 c v w x y z + 2 b^4 c v w x y z - a^2 b c^2 v w x y z +
3 b^3 c^2 v w x y z - 2 a^2 c^3 v w x y z + 4 a b c^3 v w x y z -
3 b^2 c^3 v w x y z - 2 b c^4 v w x y z + c^5 v w x y z +
a^4 c w^2 x y z - 2 a^3 b c w^2 x y z + 2 a b^3 c w^2 x y z -
b^4 c w^2 x y z - a^2 c^3 w^2 x y z + b^2 c^3 w^2 x y z +
a^5 u^2 y^2 z - a^3 b^2 u^2 y^2 z + a^3 c^2 u^2 y^2 z -
a^5 u v y^2 z + 2 a^4 b u v y^2 z + a^3 b^2 u v y^2 z -
2 a^2 b^3 u v y^2 z - 2 a^3 b c u v y^2 z + a^3 c^2 u v y^2 z +
2 a^2 b c^2 u v y^2 z - a^4 b v^2 y^2 z + b^5 v^2 y^2 z +
a^3 b c v^2 y^2 z + a b^3 c v^2 y^2 z - 2 b^3 c^2 v^2 y^2 z -
a b c^3 v^2 y^2 z + b c^4 v^2 y^2 z + 2 a^3 b c u w y^2 z -
2 a^2 b^2 c u w y^2 z - 2 a^3 c^2 u w y^2 z + 2 a^2 c^3 u w y^2 z -
a^4 b v w y^2 z + 2 a^2 b^3 v w y^2 z - b^5 v w y^2 z +
a^2 b^2 c v w y^2 z - 2 a b^3 c v w y^2 z + b^4 c v w y^2 z -
2 a^2 b c^2 v w y^2 z + 2 b^3 c^2 v w y^2 z - a^2 c^3 v w y^2 z +
2 a b c^3 v w y^2 z - 2 b^2 c^3 v w y^2 z - b c^4 v w y^2 z +
c^5 v w y^2 z - a^3 b c w^2 y^2 z + a^2 b^2 c w^2 y^2 z +
a b^3 c w^2 y^2 z - b^4 c w^2 y^2 z - a^2 c^3 w^2 y^2 z -
a b c^3 w^2 y^2 z + 2 b^2 c^3 w^2 y^2 z - c^5 w^2 y^2 z -
a^5 u^2 x z^2 - a^3 b^2 u^2 x z^2 - a^3 b c u^2 x z^2 -
a b^3 c u^2 x z^2 + 2 a^3 c^2 u^2 x z^2 + a b^2 c^2 u^2 x z^2 +
a b c^3 u^2 x z^2 - a c^4 u^2 x z^2 + 2 a^3 b^2 u v x z^2 -
2 a^2 b^3 u v x z^2 + 2 a b^3 c u v x z^2 - 2 a b^2 c^2 u v x z^2 +
a^2 b^3 v^2 x z^2 + b^5 v^2 x z^2 - b^3 c^2 v^2 x z^2 +
a^5 u w x z^2 - a^3 b^2 u w x z^2 - a^4 c u w x z^2 +
2 a^3 b c u w x z^2 - 2 a^2 b^2 c u w x z^2 - b^4 c u w x z^2 -
2 a^3 c^2 u w x z^2 + a b^2 c^2 u w x z^2 + 2 a^2 c^3 u w x z^2 -
2 a b c^3 u w x z^2 + 2 b^2 c^3 u w x z^2 + a c^4 u w x z^2 -
c^5 u w x z^2 + a^2 b^3 v w x z^2 - b^5 v w x z^2 +
2 a^2 b^2 c v w x z^2 - 2 a b^3 c v w x z^2 + 2 b^4 c v w x z^2 +
b^3 c^2 v w x z^2 - 2 b^2 c^3 v w x z^2 + a^4 c w^2 x z^2 -
a^3 b c w^2 x z^2 + a b^3 c w^2 x z^2 - b^4 c w^2 x z^2 -
2 a^2 c^3 w^2 x z^2 + a b c^3 w^2 x z^2 + c^5 w^2 x z^2 -
a^5 u^2 y z^2 - a^3 b^2 u^2 y z^2 + a^3 c^2 u^2 y z^2 +
2 a^3 b^2 u v y z^2 - 2 a^2 b^3 u v y z^2 - 2 a^3 b c u v y z^2 +
2 a^2 b c^2 u v y z^2 + a^2 b^3 v^2 y z^2 + b^5 v^2 y z^2 +
a^3 b c v^2 y z^2 + a b^3 c v^2 y z^2 - a^2 b c^2 v^2 y z^2 -
2 b^3 c^2 v^2 y z^2 - a b c^3 v^2 y z^2 + b c^4 v^2 y z^2 +
a^5 u w y z^2 - a^3 b^2 u w y z^2 - 2 a^4 c u w y z^2 +
2 a^3 b c u w y z^2 - 2 a^2 b^2 c u w y z^2 - a^3 c^2 u w y z^2 +
2 a^2 c^3 u w y z^2 + a^2 b^3 v w y z^2 - b^5 v w y z^2 +
a^4 c v w y z^2 + 2 a^2 b^2 c v w y z^2 - 2 a b^3 c v w y z^2 +
b^4 c v w y z^2 - a^2 b c^2 v w y z^2 + 2 b^3 c^2 v w y z^2 -
2 a^2 c^3 v w y z^2 + 2 a b c^3 v w y z^2 - 2 b^2 c^3 v w y z^2 -
b c^4 v w y z^2 + c^5 v w y z^2 + a^4 c w^2 y z^2 -
a^3 b c w^2 y z^2 + a b^3 c w^2 y z^2 - b^4 c w^2 y z^2 -
a b c^3 w^2 y z^2 + 2 b^2 c^3 w^2 y z^2 - c^5 w^2 y z^2 = 0.

Hyacinthos #20527

Francisco Javier García Capitán
12 December 2011

[Jean-Pierre Ehrmann]:
Suppose that P & P' are antipodes upon a rectangular hyperbola.
Consider a line L and a variable point M upon the hyperbola; L'= reflection of MP wrt L; then the reflection of MQ wrt L' goes through a fixed point when M moves upon the hyperbola (this can be shown with an easy computation. Synthetic proof?). If we take successively for M the infinite points of the asymptots, we get an easy localization of the common point.

From this, it follows that if we consider a triangle ABC, a point P, its antigonal P' and a line L. If L1,L2,L3 are the reflections of AP,BP,CP wrt L, then the reflections of AP' wrt L1, of BP' wrt L2, of CP' wrt L3 are concurrent (and if, instead of ABC, we take any triangle inscribed in the rectangular circumhyperbola going through P, the common point will be the same one)
For instance, if P = I, then P'=X[80]

Hyacinthos 20538