Consider any convex quadrilateral 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>Q</mi></mrow></math> made from four rigid rods

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with flexible joints at the vertices so that the shape of 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>Q</mi></mrow></math> can

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be changed while keeping the lengths of the sides constant. If the

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diagonals of the quadrilateral cross at an angle 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>&#x03B8;</mi></mrow></math> in the

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range 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mo stretchy="false">(</mo><mn>0</mn><mo>&#x2266;</mo><mi>&#x03B8;</mi><mo>&#x003C;</mo><mi>&#x03C0;</mi><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></math>, as we deform 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>Q</mi></mrow></math>, the angle

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<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>&#x03B8;</mi></mrow></math> and the lengths of the diagonals will change. Using the

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results of the two problems on quadrilaterals in the October 2002

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15+ Challenges prove that the area of 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>Q</mi></mrow></math> is proportional to

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<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>tan</mi><mi>&#x03B8;</mi></mrow></math>.

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