In the parallelogram ABCD, it is known that the lengths of two diagonal lines AC and BD are 4 and 6 respectively, then AB & # 178; + BC & # 178=

In the parallelogram ABCD, it is known that the lengths of two diagonal lines AC and BD are 4 and 6 respectively, then AB & # 178; + BC & # 178=

Note: AC is a long diagonal, which is proved by Pythagorean theorem. Proof: from the parallelogram ABCD, we know that ad = BC, DC is parallel to AB, through D for DN, vertical to AB in N, through C for cm, vertical to ab extension in M, then DN = cm, so RT triangle adn and RT triangle BCM are congruent (HL theorem), that is, an = BM, so in RT triangle adn and RT triangle BCM, an2 + DN2 = BM2 + cm2 = ad2 = BC2, BD2 = DN2 + BN2 = DN2 + (ab-an) 2 = DN2 + an2-2an * AB + AB2 = ad2-2an * AB + AB2; in RT triangle ACM, ac2 = BC2 + 2bm * AB + AB2 = ad2 + 2An * AB + AB2; so ac2 + BD2 = ad2-2an * AB + AB2 + ad2 + 2An * AB + AB2 = 2 (AB2 + ad2)
Because ABCD is a parallelogram
So ad = BC
So AB2 + BC2 = (ac2 + BD2) / 2 = 2 √ 13