Geometry · Rectangle · Triangle area · Extremal position · Half square triangle
A rectangle \(ABCD\) has \(|AB| = 2a\) and \(|AD| = a\). Let \(E\) be the midpoint of the side \(AB\), and choose an arbitrary point \(F\) on the side \(AD\). The area of the triangle \(ECF\) depends on where \(F\) is taken. What is the smallest and what is the largest area the triangle \(ECF\) can have?
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), izbirno (selection) round 2004, 1. letnik, category A, problem 3. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source