Problem
GEO-B1-M05-P027 Parallelism of a Tangent and a Chord
In triangle \(ABC\), the tangent to the circumcircle at \(A\) meets the line through \(C\) parallel to \(AB\) at point \(T\). Find a necessary and sufficient condition for \(AT=CT\).
Use \(CT\parallel AB\), then compare the angles at \(A\) and \(C\) in triangle \(ACT\).
Since \(CT\parallel AB\), angle \(\angle ACT\) equals the angle between \(AC\) and \(AB\), that is \(\angle CAB\). By the tangent-chord theorem, \(\angle TAC=\angle ABC\). In triangle \(ACT\), equality \(AT=CT\) is equivalent to equality of base angles: \(\angle ACT=\angle TAC\). Hence \(AT=CT\) if and only if \(\angle CAB=\angle ABC\), that is, if and only if \(AC=BC\).
A hard “if and only if” problem: the tangent and the parallel line translate a segment equality into an angle equality in the original triangle.