Problem
GEO-B2-M01-P014 Tangent Circles and Parallel Chords
#14
★★★★☆ Level 4 of 5
Two circles are tangent at \(A\). Two lines through \(A\) meet the first circle again at \(B\) and \(C\), and the second circle again at \(D\) and \(E\), respectively. Prove that \(BC\parallel DE\).
Use the common tangent at \(A\) and the tangent-chord theorem for both circles.
Let \(t\) be the common tangent at \(A\). For the first circle, \(\angle(t,AB)\equiv\angle ACB\). For the second circle, since \(AD\) lies on the same line as \(AB\), we have \(\angle(t,AD)\equiv\angle AED\). Hence \(\angle ACB\equiv\angle AED\). Since \(A,C,E\) are collinear, this means that lines \(BC\) and \(DE\) form equal angles with line \(CE\). Therefore \(BC\parallel DE\).
A strong but clean tangent problem: no lengths, only angles.