- Dear Jean-Pierre

I will look at your solution very carefully.

At first sight, it looks good for you use only affine objects and the theory

is affine!

Of course my construction maybe slightly different for I only look at the

affine map ABC --> a(t)b(t)c(t) and don't think to study the affine map

a(t)b(t)c(t) --> a(t')b(t')c(t') even if I have often used it in other

situations. That's why I said your remark was so interesting and effectively

leads to a new construction.

Your remark means simply that this affine map a(t)b(t)c(t) -->

a(t')b(t')c(t') is the product of a translation and a transvection.

In fact all the choo-choo theory is based on the classification of the

affine plane maps and especially about their fixed points and invariant

lines ( or axis as you said it).

When you think that our students in Math Agregation are supposed to know the

decomposition of linear endomorphisms in any dimension over any field and

are unable to apply this theory in the affine plane case over the real

field, one must be sad! Teachers knowing invariants of a matrix and beeing

unable to do a simple plane drawing, I cry!

Just some final remarks about this area duck problem:

1° Of course this is a 2th degree problem over the real field, so there are

2, 1, 0 or infinitely many solutions! That's just the pleasure of the

discussion.

2° The support lines of the duck's motions are not needed to form a triangle

ABC even if as Hyacinthists we are only in love with these damned "

Donalded" triangles!

Friendly

Francois

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__[APH]:__Let ABC be a triangle.

A line L passing through A intersects the circle with diameter AB again at Ab and the circle with diameter AC again at Ac.

[Equivalently: Let Ab, Ac be the orthogonal projections of B, C on L, resp.]

Let A* be the intersection of BAc and CAb.Which is the locus of A* as L moves around A?

[César Lozada]:

Parametric trilinear equation:

1/u(t) = a*((b^2+c^2-a^2)^2-4*b^2*c^2*c os(2*t)^2)/(2*S)

1/v(t) = 2*(cos(2*t)*c-b)*S - c*sin(2*t)*(a^2+3*b^2-2*cos(2* t)*b*c-c^2)

1/w(t) = 2*(cos(2*t)*b-c)*S + b*sin(2*t)*(a^2+3*c^2-2*cos(2* t)*b*c-b^2)

Regards,

César Lozada