Question

求证 ：平行四边形的两条对角线长度的平方和等于四边长度的平方和,

Answer 1

AC^2=AB^2+BC^2-2AB*BCcosB

BD^2=BC^2+CD^2-2BC*CD*cosC

AC^2+BD^2=AB^2+BC^2-2AB*BCcosB + BC^2+CD^2-2BC*CD*cosC

= AB^2+BC^2+ BC^2+CD^2- 2BC(ABcosB +CD*cosC)

∵AB=CD,BC=AD,∠B+∠C=180°

∴AC^2+BD^2= AB^2+BC^2+ AD^2+CD^2- 2BC(ABcosB +ABcos(180°-B))

= AB^2+BC^2+ AD^2+CD^2- 2BC(ABcosB -ABcosB)

= AB^2+BC^2+ AD^2+CD^2,得证