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We can prove this by using Pythagoras theorem.  At B we have a right angle.

In the triangle, ABE,  we get:  AE² = AB² + BE²

In the triangle  DBC, we get :  CD² = DB² + BC²

adding the above two equations. we get:
    AE² + CD² = (AB² + BC²) +( DB² + BE²)
                   =  AC²  +  DE²

This equation will be true for a quadrilateral with two adjacent sides being perpendicular.

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