Re: [EMHL] Re: Locus
- 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
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
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 "
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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?
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)