Problem
GEO-B2-M10-P018 Two Tangents and a Secant
#18
★★★★☆ Level 4 of 5
The tangents to a circle at \(A\) and \(C\) meet at \(T\). A line through \(T\) meets the circle at \(B\) and \(D\). Prove that \(TA=TC\) and \(TB\cdot TD=TA^2\).
Use equality of tangents from one point and the power of point \(T\).
Tangents drawn from one point to a circle are equal, so \(TA=TC\). Also, the power of point \(T\) with respect to the circle equals both \(TA^2\) and \(TB\cdot TD\) along secant \(TBD\). Therefore \(TB\cdot TD=TA^2\).
The problem shows that one point can give several expressions for the same power.