RU-ALLRUSSIAN 1966 Problem 2

(a) B and C are on the segment AD with $\mathrm{AB} = \mathrm{CD}$ . Prove that for any point P in the plane: $\mathrm{PA} + \mathrm{PD} \geq \mathrm{PB} + \mathrm{PC}$ .

(b) Given four points A, B, C, D on the plane such that for any point P on the plane we have $\mathrm{PA} + \mathrm{PD} \geq \mathrm{PB} + \mathrm{PC}$ . Prove that B and C are on the segment AD with $\mathrm{AB} = \mathrm{CD}$ .