Hatcher-Topology-of-Numbers · Problem 1.2.3

Continued Fractions

Mathematics — Algebraic Topology · ford-circles-point-of-tangency

Problem

Suppose two Ford circles tangent to the

x

-axis at points

a/b

and

c/d

are tangent to each other. Show that the point of tangency between the two circles is the point

\left(\frac{ab + cd}{b^2 + d^2},\, \frac{1}{b^2 + d^2}\right)

so in particular the coordinates of this point are rational. Hint: What proportion of the way along the line segment joining the two centers is the point of tangency? This same proportion will apply to

x

-coordinates and

y

-coordinates separately.

Related theory

The Equivalence Point And The End Point

Browse the problem library