Problem
GEO-B1-M08-P029 Two Tangents
#29
★★★★★ Level 5 of 5
Tangents to the circumcircle of triangle \(ABC\) at points \(B\) and \(C\) meet at point \(T\). Prove that \(\angle BTC=180^\circ-2\angle BAC\).
Connect the centre of the circle to \(B\) and \(C\).
Let \(O\) be the centre of the circle. Radii \(OB\) and \(OC\) are perpendicular to the tangents, so in quadrilateral \(BOCT\), the angles at \(B\) and \(C\) are right. The central angle \(\angle BOC=2\angle BAC\). Hence \(\angle BTC=360^\circ-90^\circ-90^\circ-2\angle BAC=180^\circ-2\angle BAC\).
A strong mixed problem: tangents, radii, central and inscribed angles.