Problem
GEO-B2-M01-P003 Tangent and Chord
#3
★★☆☆☆ Level 2 of 5
Line \(t\) is tangent to the circumcircle of triangle \(ABC\) at \(A\). Prove that \(\angle(t,AB)\equiv\angle ACB\pmod{180^\circ}\).
This is the tangent-chord theorem. It can be proved using the radius to the point of tangency and a central angle.
Let \(O\) be the centre of the circle. Radius \(OA\) is perpendicular to tangent \(t\). The central angle \(\angle AOB\) is twice the inscribed angle \(\angle ACB\). From \(OA\perp t\), the angle between \(t\) and \(AB\) equals half of the central angle, that is \(\angle ACB\). Hence \(\angle(t,AB)\equiv\angle ACB\pmod{180^\circ}\).
This is a core tool of the module; later it is used as a ready fact.