Problem
GEO-B2-M01-P011 Two Tangents to a Circle
#11
★★★☆☆ Level 3 of 5
Tangents to the circumcircle of triangle \(ABC\), where \(\angle BAC<90^\circ\), are drawn at \(B\) and \(C\), meeting at \(T\). Prove that the smaller angle between the tangents equals \(180^\circ-2\angle BAC\).
Join the centre of the circle to points \(B\) and \(C\).
Let \(O\) be the centre of the circle. Then \(OB\perp TB\) and \(OC\perp TC\). Since \(\angle BAC<90^\circ\), the smaller central angle \(\angle BOC\) standing on arc \(BC\) equals \(2\angle BAC\). The smaller angle between the tangents is supplementary to this central angle. Therefore it equals \(180^\circ-2\angle BAC\).
In examples it is important to distinguish the smaller angle between tangents from the exterior oriented angle.