Consider ΔABC.
Given that
DCBD=AFBFConverse of Triangle Proportionality Theorem
If a line divides two sides of a triangle proportionally then it is parallel to the third side.
Hence FD∥AC
∠FDA=∠CAD
Consider a convex quadrilateral AFKE.
∠BFC+∠AFC=180∘ Given that ∠ADB=∠AFC. Then
∠BFC+∠ADB=180∘
∠BFK+∠KDB=180∘The measure of the interior angles of a convex quadrilateral is the same as the sum of the measures of the interior angles of two triangles, or 360 degrees.
∠BFK+∠KDB=180∘=∠FBD+∠FKD If two opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.
From Inscribed Angle Theorem we have
∠FBK=∠FDK
∠ABE=∠FDA Therefore,
∠ABE=∠FDA=∠CAD Therrefore,
∠ABE=∠CAD
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