Seminar: Week 2

A rain of implications  (Unbearable lightness of the derivations...)

1.  Conjugated disks :  $D + D' =2(A+B+C)$.
2.  (Generalized) Pappus' chains:  $C_{n+1}= 2C_n - C_{n-1} + 2(A+B)$.
3.  Pappus thread:  $K=\pm \frac{A-B}{2}$.
4.  Mid-circle:  $T= \pm \frac{A+B+C-D}{2}$.
5.  Points of tangency:  $P=\frac{A+B}{2}$. 

Seminar: Week 1

1.  Introduction.  Plan for the near future.  Chart of the ideas (see below).

2.  The map $\mathcal D_2 \ \to \ \mathbb M^{1,3}$ that is central for this topic.

Intro

For the list of relevant papers see

>>>  this page (click) <<<

For some varia (including applets)

>>> go here <<<

Challenge:  Prove the following

If a quadruple of four non-negative integers $A$, $B$, $C$, $D$ is primitive, i.e., $\gcd(A,B,C,D)=1$, satisfies $$2(A^2 + B^2 + C^2 + D^2) = (A+B+C+D)^2$$ then any two add up to a sum of two squares: $$A+B = p^2+q^2$$ for some integers $p$ and $q$.