20 13 China normal university recruitment second question. The radius r of the circumscribed circle of the triangle abc and the distance d from the center of gravity to the outer center prove that a2+b2+c2+d2=9R2. Wrong topic? I think it is concerned about the outside world.

The first reaction to this question is calculation? Calculate the relationship between D and triangle, and substitute it into a2+b2+c2+d2 test, and you will find that D is actually not beautiful (because I use the three-point * * line with the ratio of the center of gravity and 1: 2 to determine D, and the distance between the center of gravity and the center of the heart will directly take an edge as a vertical parallel line and then make the Pythagorean theorem).

Another idea is vectors,

Let the outer center O of ABC be perpendicular to H, then OH=OA+OB+OC (here all refer to quantities? The same below) (This can be verified by directly calculating (OH-OA)*(OB-OC)=0. In other words, you can make such an H before he is interested)? So AB? +AC？ +BC？ +Oh? =(OA-OB)？ +(OA-OC)？ +(OB-OC)？ +(OA+OB+OC)？ =3(OA？ +OB？ +OC？ )=9R？