varianta 1
aria ΔAMC=AC*PM/2⇒MP=2 ArieΔAMC/AC
arieΔAMB=AB*MN/2⇒MN=2ArieΔAMB/AB
MP+MN= 2 ( ArieΔ AMC/AC + ArieΔAMB/AB)
dar AB=AC
deci
MP+MN= (2/AB) *( ArieΔ AMC+ArieΔAMB)
=(2/AB) *Arie ΔABC
(2/AB)* AB²/2=AB=constant
Altfel , varianta 2
MN⊥AB, mas∡ B=45°⇒ΔMNB dreptunghic isoscel⇒MN≡NB 91)
mas ∡P=mas∡A=mas∡N=90°, PAMN patrulater convezx, suma mas.unghiurilor 360°⇒mas∡ PMN=90°⇒PAMN dreptunghi⇒PM≡AN (2)
din (1) si (2)⇒MP+MN=AN+NB=AB =constant, cerinta
