Как решаются такие задачи?

Question

I don’t understand how to solve such problems. That is, I do not understand what to do in them. Explain, please.

First task: Prove that if A, B, C, D are four arbitrary points (in a plane or in space), and the point P is the middle of the segment AC and the point Q is the middle of the segment BD, then (everything will be vectors below) $$| AB | ^ 2 + | BC | ^ 2 + | CD | ^ 2 + | DA | ^ 2 = | AC | ^ 2 + | BD | ^ 2 + 4 | PQ | ^ 2$$

The second task: Prove the identity: (here, too, all the letters of the vector) $$(a + b, b + c, c + a) ^ 2 =2 (a, b, c).$$ Here, especially here, what needs to be done here, how to prove this identity?

Photo:

Answer 1

Guide for the first task:

• Use that $|\vec{XY}|^2= \vec{XY}\cdot \vec{XY}$
• $\vec{XY} = Y-X$
• If $M$ is a midpoint of $XY$ then $M = {1\over 2}(X+Y)$

Everything is just use of exactly this formulas!