Puzzles
Draw one circle that touches three others — how many ways can it be done?
You're given three circles of any size, placed anywhere on a page. Using only a compass and straightedge, construct a fourth circle tangent to all three — touching each one without crossing it. More than one such circle usually exists. How many are there for a typical arrangement, and can you actually construct even one?
Reveal the answer
Up to eight solution circles exist for a generic arrangement of three circles — one for each way of splitting them into groups touched from outside versus inside. Apollonius of Perga solved the problem around 200 BC in his lost treatise Tangencies, known today only through a 4th-century summary by Pappus of Alexandria; François Viète found a full compass-and-straightedge construction in the 16th century.