Problem
GEO-B1-M05-P025 Converse Tangent Theorem
#25
★★★★☆ Level 4 of 5
Through point \(B\) of triangle \(ABC\), a line \(l\) outside the triangle is drawn. It is known that the angle between \(l\) and chord \(BA\) equals \(\angle BCA\). Prove that \(l\) is tangent to the circumcircle of triangle \(ABC\) at point \(B\).
Recall the converse form of the tangent-chord theorem.
For the circle through \(A,B,C\), the tangent at \(B\) forms with chord \(BA\) an angle equal to the inscribed angle \(\angle BCA\), which stands on chord \(BA\). By condition, line \(l\) forms exactly this angle with \(BA\) and lies outside the triangle. Therefore \(l\) is the tangent to the circumcircle at point \(B\).
A strong problem on the converse use of the tangent-chord theorem.