Problem
GEO-B2-M02-P018 A Tangent to a Small Circle
#18
★★★★☆ Level 4 of 5
In triangle \(ABC\), point \(D\) lies on side \(BC\). From point \(C\), tangent \(CE\) is drawn to circle \((ABD)\), where \(E\) is the point of tangency. Prove that \(CE^2=CB\cdot CD\).
For circle \((ABD)\), line \(CBD\) is a secant from point \(C\).
With respect to circle \((ABD)\), from point \(C\) we have tangent \(CE\) and secant \(CDB\). Since \(D\) lies on side \(BC\), the points on the secant are in the order \(C,D,B\). By the tangent-secant formula, \(CE^2=CD\cdot CB\).
The problem teaches students to see power of a point with respect to a small circle inside a triangle.