G. Polya "Mathematics and plausible reasoning" Chapter 9, problem 2:
Three circles in a plane, exterior to each other, are given in position. Find the triangle with minimum perimeter that has one vertex on each circle.
From the contents of the chapter it is obvious (using light reflections on three circular mirrors and rubber band methods) that the two sides of the required triangle that meet in a vertex on a given circle include equal angles with the radius.
But how to construct (with the compass and straightedge) these vertices (A,B,C)?
Let one of the circles be an infinite radius (a straight line):
Looks like the same solution... And no idea about construction.
So let all of the circles be an infinite radius:
And we get Fagnano's problem with clear construction.
Hope this will be useful (?)