Hatcher-Topology-of-Numbers · Problem 1.1.3

Continued Fractions

Mathematics — Algebraic Topology · farey-diagram-mediant-triangle

Problem

a/b

c/d

, and

e/f

are fractions in lowest terms such that

e/f

is the mediant of

a/b

and

c/d

, is it necessarily true that there is a triangle in the Farey diagram with vertices

a/b

c/d

, and

e/f

? Give either a proof or a counterexample.