**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: $P=\frac{A+B}{2}$.

# Apolloniana

On Apollonian and other disk arrangements:

geometry, number theory, physics

## Thursday, September 14, 2023

### Seminar: Week 2

## Thursday, September 7, 2023

### 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.

## Tuesday, October 27, 2020

### Intro

For the list of relevant papers see

>>>

**this page (click) <<<**

**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 $q$.

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