Problem
GEO-B1-M05-P029 Tangents at Two Vertices of a Triangle
#29
★★★★★ Level 5 of 5
In triangle \(ABC\), the tangents to the circumcircle at points \(B\) and \(C\) meet at point \(T\). Prove that \(\angle BTC=180^\circ-2\angle BAC\).
Connect the centre \(O\) of the circle to points \(B\) and \(C\).
Let \(O\) be the circumcentre. Radii \(OB\) and \(OC\) are perpendicular to tangents \(TB\) and \(TC\), so \(\angle OBT=\angle OCT=90^\circ\). The central angle \(\angle BOC\) standing on arc \(BC\) equals \(2\angle BAC\). In quadrilateral \(BOCT\), the angle sum is \(360^\circ\). Hence \(\angle BTC=360^\circ-90^\circ-90^\circ-\angle BOC=180^\circ-2\angle BAC\).
A strong first-level problem: tangents, radii, and central angles are combined.