MathDB

3

Part of 1999 German National Olympiad

Problems(1)

A =\frac{a+c}{2\frac{b+d}{2} , A = \sqrt{(p-a)(p-b)(p-c)(p-d)} , area quadril.

Source: Germany 1999 p3

2/23/2020
A mathematician investigates methods of finding area of a convex quadrilateral obtains the following formula for the area AA of a quadrilateral with consecutive sides a,b,c,da,b,c,d: A=a+c2b+d2A =\frac{a+c}{2}\frac{b+d}{2} (1) and A=(p−a)(p−b)(p−c)(p−d)A = \sqrt{(p-a)(p-b)(p-c)(p-d)} (2) where p=(a+b+c+d)/2p = (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.
geometryrectangleCyclicarea