Problem
GEO-B2-M01-P023 Two Tangents at One Point
#23
★★★★★ Level 5 of 5
In triangle \(ABC\), point \(D\) lies on \(BC\). Tangents are drawn at \(D\) to circles \((ABD)\) and \((ACD)\). It is known that these tangents are perpendicular. Prove that \(\angle BAC=90^\circ\).
Use the fact that the angle between circles \((ABD)\) and \((ACD)\) at \(D\) equals \(\angle BAC\).
The angle between two circles at \(D\) is the angle between their tangents at that point. By the angle fact already proved for \(D\in BC\), the angle between circles \((ABD)\) and \((ACD)\) equals \(\angle BAC\) in oriented notation. By the condition, the tangents are perpendicular, so this angle is \(90^\circ\). Therefore \(\angle BAC=90^\circ\).
A short strong problem: it checks whether the student can use an intermediate fact as a tool.