Problem
GEO-B2-M01-P022 A Tangency Criterion Through Parallelism
Circles \((ABC)\) and \((ADE)\) have common point \(A\), with points \(B,A,D\) collinear and points \(C,A,E\) collinear. Prove that these circles are tangent at \(A\) if and only if \(BC\parallel DE\).
Compare the angles between the common line \(AB/AD\) and the tangents at \(A\), then translate this into angles \(ACB\) and \(AED\).
Let \(t_1\) and \(t_2\) be the tangents to circles \((ABC)\) and \((ADE)\) at \(A\). By the tangent-chord theorem, \(\angle(t_1,AB)\equiv\angle ACB\), and \(\angle(t_2,AD)\equiv\angle AED\). Since \(AB\) and \(AD\) are one line, the tangents coincide if and only if \(\angle ACB\equiv\angle AED\). But points \(A,C,E\) are collinear, so this angle equality is equivalent to \(BC\parallel DE\). Therefore the circles are tangent at \(A\) if and only if \(BC\parallel DE\).
An excellent closing problem for the topic: tangency, chord, parallelism, and oriented angles work together.