Problem
GEO-B2-M01-P018 A Tangent Meets a Side
#18
★★★★☆ Level 4 of 5
The tangent to the circumcircle of triangle \(ABC\) at \(A\) meets line \(BC\) at \(T\). Prove that \(\triangle TAB\sim\triangle TCA\).
Two angles come from the tangent-chord theorem, and two more from the collinearity of \(T,B,C\).
By the tangent-chord theorem, \(\angle TAB\equiv\angle ACB\). Since \(T,C,B\) are collinear, \(\angle ACB=\angle TCA\). Hence \(\angle TAB=\angle TCA\). Similarly, \(\angle TAC\equiv\angle ABC=\angle TBA\). Therefore \(\triangle TAB\sim\triangle TCA\).
The problem looks like similarity, but the key to it is tangent and chord.