Triangular orbits in elliptic billiards - Triangle geometry
I made quicky current public worsheet to share about some of my computations.
Parametrization in triangle geometry
I got following parametrization of a 3-orbit triangle ABC in the circumbilliard :
with parameters as ratio of inradius and circumradius (), as cosine of internal angle at A vertex, and as semiperimeter.
It comes from a Ravi substitution (, , ) and three equations in x,y,z unknowns given parameters t,s,k :
Solving is best (simplest) for the couple of first and third equations, expressing y,z from x.
Putting back expression into second equation we have the binomial equation :
where sum of roots is
and product is
Choosing one root or the other is only changing sign of w (or swapping b and c).
Circle centered at Mittenpunk and with radius is the cosine circle for the excentral triangle T.
The excentral triangle T has for vertices, , the excenters of the reference ABC triangle.
Its symmetrical image T' wrt Mittenpunkt has vertices , and .
Mittenpunkt has trilinear coordinates : .
Triangle T and its symmetrical image T' intersect at six points which are on the cosine circle.
Trilinear coordinates of these points are :
Computing square of quadrance (=squared distance) between and we get squared radius of cosine circle :
and using equations for and :
Parallels with respect to ABC edges through , cut disk in six sectors, and cutting points on cosine circle are intersection of T and T' edges.
Squared cosines of internal angles of external triangle T are :
Product : .
Sum of squared cosines is : .
Sum of product 2 by 2 of squared cosines is : where is the changing squared inradius of ABC : .
Exradii are :