Problem
GEO-B2-M02-P009 A Circle Inside a Triangle
#9
★★★☆☆ Level 3 of 5
In triangle \(ABC\), points \(D\) and \(E\) lie on sides \(AB\) and \(AC\). It is known that \(B,C,D,E\) lie on one circle, \(AD=4\), \(AB=14\), \(AE=7\). Find \(AC\).
Consider the power of point \(A\) with respect to circle \((BCDE)\).
Lines \(AB\) and \(AC\) are secants of circle \((BCDE)\). Therefore \(AD\cdot AB=AE\cdot AC\). We get \(4\cdot14=7\cdot AC\), hence \(AC=8\).
The problem shows how power of a point appears without an external point outside the whole figure.