I am not a specialist in maths, so I thank you very much for any help you can give me.


Consider two circles C1, C2.


Q1: Find the points that are in the intersection of C1 and C2, this is easy !


Q2: Find two points p1 and p2, such that (p1 \in C1) and (p2 \in C2), and (distance(p1, p2)= D).
Is it possible to solve this problem ?


Now I want to generalize it to more than two circles and to arbitrary geometric predicates (or patterns)


Consider the circles C1, C2, ..., Cn


Q3: Find the points (p1 \in C1), (p2 \in C2), ... , (pn \in Cn) such that (Some\_Geometric\_Predicate(p1, ..., pn)= true).


Have you encountered this problem before ? are they any references that speak about such kind of problems ?


Thank you in advance for your help.



Points in circles that form a given geometric pattern
Permit me to reformulate a specific version of Q3 that
Ellipsissi posed in the comments:



> 
> **P1**. Given three non-intersecting circles
>  $\{C\_1,C\_2,C\_3\}$,
>  find all triples
>  $\{p\_1,p\_2,p\_3\}$ with $p\_i \in C\_i$
>  such that $\triangle p\_1 p\_2 p\_3$ is
>  similar to a given triangle $T$.
> 
> 
> 


This differs from the posed question in (a) the non-intersecting condition,
and (b) not demanding that a specific angle be realized at
a specific corner $p\_i$.
The form above is analogous to this problem:



> 
> **P2**. Given a plane curve $\gamma$,
>  find all triples
>  $\{p\_1,p\_2,p\_3\}$ of points on $\gamma$
>  such that $\triangle p\_1 p\_2 p\_3$ is
>  similar to a given triangle $T$.
> 
> 
> 


Much is known about P2, under various restrictions on $\gamma$.
For example, if $\gamma$ is a smooth Jordan curve,
then I believe it is almost completely understood now,
through recent work of
Benjamin Matschke, and of
Jason Cantarella, Elizabeth Denne, and John McCleary.
See especially Cantarella's fascinating [web pages on the topic](http://www.jasoncantarella.com/webpage/index.php?title=Square_Peg_problem#Results_on_Inscribed_Triangles).


So, here is a high-level plan for P1. Connect the three circles by thin
corridors to form a plane curve $\gamma$. 
Solve P2, and discard solutions with points on the corridors, or more than one point
on one $C\_i$.
The efficacy of this plan depends on the degree to which
P2 is completely solved in its various guises.


**References**


1.
M. J. Nielsen. "Triangles inscribed in simple closed curves," 
*Geometriae Dedicata* **43**: 291-297 (1992).


2.
Benjamin Matschke.
"On the Square Peg Problem and some Relatives."
[arXiv](http://arxiv.org/abs/1001.0186) (2009)


3.
Wikipedia article on the [Inscribed Square Problem](http://en.wikipedia.org/wiki/Inscribed_square_problem), with triangles discussed under "Variants."


