MathDB

A mathematician investigates methods of finding area of a convex quadrilateral obtains the following formula for the area

A

of a quadrilateral with consecutive sides

a,b,c,d

A =\frac{a+c}{2}\frac{b+d}{2}

(1) and

A = \sqrt{(p-a)(p-b)(p-c)(p-d)}

(2) where

p = (a+b+c+d)/2

. However, these formulas are not valid for all convex quadrilaterals. Prove that (1) holds if and only if the quadrilateral is a rectangle, while (2) holds if and only if the quadrilateral is cyclic.

Problem Statement