Question

如图所示,AD是△ABC的中线.试判断AB²;+AC²;=2(AD²;+DC²)

Answer 1

做AE垂直于BC于E,则有

AB^2=BE^2+AE^2

AC^2=CE^2+AE^2

AE^2=AD^2-DE^2

BD=CD

BE=BD+DE=CD+DE

CE=CD-DE

AB^2+AC^2=(CD+DE)^2+(CD-DE)^2+2*(AD^2-DE^2)

=2*CD^2+2*DE^2+2*AD^2-2*DE^2

=2(AD^2+DC^2)