Problem
GEO-B2-M01-P020 A Tangent Parallel to a Side
#20
★★★★☆ Level 4 of 5
In triangle \(ABC\), point \(D\) lies on side \(BC\). The tangent to circle \((ABD)\) at \(D\) is parallel to \(AC\). Prove that \(\angle BAC=\angle ACB\).
Compare the angle between the tangent and \(DB\) with angle \(DAB\).
By the tangent-chord theorem, the angle between the tangent at \(D\) and chord \(DB\) equals \(\angle DAB\). Since the tangent is parallel to \(AC\), and \(DB\) lies on \(BC\), the same angle equals \(\angle ACB\). Also \(D\) lies on \(BC\), so \(\angle DAB=\angle BAC\). Therefore \(\angle BAC=\angle ACB\).
The problem teaches students to see a tangent to a smaller circle, not only to the whole triangle's circumcircle.