MathDB

6

Part of 2005 Postal Coaching

Problems(1)

trapezoid, prove two equations

Source: National round in Slovakia 2004/2005, problem 3; contest 1996

4/10/2005
Let ABCDABCD be a trapezoid such that ABAB is parallel to CDCD, and let EE be the midpoint of its side BCBC. Suppose we can inscribe a circle into the quadrilateral ABEDABED, and that we can inscribe a circle into the quadrilateral AECDAECD. Denote ∣AB∣=a|AB|=a, ∣BC∣=b|BC|=b, ∣CD∣=c|CD|=c, ∣DA∣=d|DA|=d. Prove that a+c=b3+d;a+c=\frac{b}{3}+d; 1a+1c=3b\frac{1}{a}+\frac{1}{c}=\frac{3}{b}
geometrytrapezoidratiorectangleparallelogramgeometric transformationhomothety