ABC is an isosceles triangle a circle is such that it parses though vertex C and AB acts as a tangent at D for the same circle AC and BC intersects the circle at E and F respectively AC = BC = 4 cm. and AB = 6 cm also D is the mid-point of AB. What is the ratio of EC : (AE+ FE) ?

In this geometry problem, ABC is isosceles with AC = BC = 4 cm and AB = 6 cm. A circle passes through C and is tangent to AB at D (midpoint). Using power of point A, we get AE × AC = AD² = 9. Similarly, power of point B gives BF × BC = BD² = 9. Since AC = BC = 4, we get AE = BF = 9/4 = 2.25. Then EC = AC - AE = 4 - 2.25 = 1.75. Using circle properties, FE can be calculated, and the ratio EC : (AE + FE) = 14/39.