中線定理
A M 2 = 1 4 ‖ b − a ‖ 2 = 1 4 ( ‖ a ‖ 2 − 2 ⟨ a , b ⟩ + ‖ b ‖ 2 ) . {\displaystyle AM^{2}={\frac {1}{4}}\|{\boldsymbol {b}}-{\boldsymbol {a}}\|^{2}={\frac {1}{4}}(\|{\boldsymbol {a}}\|^{2}-2\langle {\boldsymbol {a}},{\boldsymbol {b}}\rangle +\|{\boldsymbol {b}}\|^{2}).}
これより、辺々を加えて2倍すると、
2 ( O M 2 + A M 2 ) = ‖ a ‖ 2 + ‖ b ‖ 2 = O A 2 + O B 2 . {\displaystyle 2(OM^{2}+AM^{2})=\|{\boldsymbol {a}}\|^{2}+\|{\boldsymbol {b}}\|^{2}=OA^{2}+OB^{2}.} Q.E.D.