Quadrilateral Inequality involving Perimeter & Diagonals

For a convex quadrilateral ABCD in the plane, the following inequality holds: 1/2 perimeter ABCD < AC + BD < perimeter ABCD.

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Quadrilateral Inequality involving Perimeter & Diagonals

Can you prove the result? If stuck, click the Hint in the sketch.

Can you provide counter-examples for the result when ABCD is concave or crossed?

(The problem was used in the 3rd Round of the South African Mathematics Olympiad (SAMO) in 2001 as the first 'warm-up' problem. A solution is given at SAMO Solutions: 2001. Past SAMO papers & solutions for several years are available at Past SAMO Papers & Solutions.)

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Michael de Villiers, Sept 2009.