Step 2: Construct a figure by the given data


             In \triangle ABC, XY || AC

            Area of \triangle BXY = Area of quadrilateral XYCA

           Area   \triangle ABC = 2 * area ( \triangle BXY)

               Area \triangle ABC = 2 * area Δ BXY.....................................(1)

Step 2:  Prove that the triangles (ΔABC, ΔBXY) are similar

             EXPLANATION:  From the figure XY || AC and BA is a transversal.

                      So, we can write \angle BXY = \angle BAC   

                         \angle XBY = \angle ABC (common angle)

                     Hence,  ΔBAC ~ ΔBXY (AAA similarity)

Step 3: Find the required ratio, by finding the areas ratio of the similar triangles.

            THEOREM: \frac{area \triangle BAC}{area \triangle BXY} = (\frac{BA}{BX})^2

                         BA = \sqrt{2} BX                  [since equation (1)]

                         BA = \sqrt{2}(BA - AX)

                         (\sqrt{2} - 1) BA = \sqrt{2} AX

                         \frac{AX}{XB} = \frac{(\sqrt{2} - 1)}{\sqrt{2}}