Problem
GEO-B2-M01-P006 Recognising a Tangent
#6
★★★☆☆ Level 3 of 5
Points \(A,B,C\) lie on a circle. Line \(l\) passes through \(A\), and \(\angle(l,AB)\equiv\angle ACB\pmod{180^\circ}\). Prove that \(l\) is tangent to the circle at \(A\).
Compare \(l\) with the actual tangent at \(A\).
Let \(t\) be the tangent to the circle at \(A\). By the tangent-chord theorem, \(\angle(t,AB)\equiv\angle ACB\pmod{180^\circ}\). By the condition, \(\angle(l,AB)\) has the same value. Therefore lines \(l\) and \(t\) coincide, and \(l\) is tangent.
It is useful to train the reverse use of the tangent-chord theorem.