Try this beautiful Problem on Geometry from Trapezium from (AMC 10 A, 2009).
Trapezium – AMC-10A, 2009- Problem 23
Convex quadrilateral $A B C D$ has $A B=9$ and $C D=12$. Diagonals $A C$ and $B D$ intersect at $E, A C=14$, and $\triangle A E D$ and $\triangle B E C$ have equal areas. What is $A E ?$
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