Problem
GEO-B2-M10-P006 Tangent to the Circumcircle
#6
★★★☆☆ Level 3 of 5
In triangle \(ABC\), the tangent to the circumcircle at \(A\) meets line \(BC\) at \(T\). Prove in directed lengths that \(TA^2=TB\cdot TC\).
Consider the power of point \(T\) with respect to the circumcircle.
From point \(T\), tangent \(TA\) and secant \(TBC\) are drawn to the circle. Hence by power of a point, \(TA^2=TB\cdot TC\) in directed lengths.
It is important that \(T\) may lie outside segment \(BC\), so directed lengths are convenient.