Apolonius Theorem Proof

Apolonius Theorem :-

Here AD is median of Triangle ABC. 

Acc. to Apolonius Theorem  AB²  + AC² =2(AD² + BD² )

PROOF:-

  1. Draw perpendicular AE to Side BC

In        ABE :-

AB² = BE² + AE²                  (1)

In        AED :-

AD² = ED² + AE²                 (2)

Now subtracting eq 2 from eq 1:-

AB²- AD²= BE² + AE² - ED² - AE²   

AB² = BE² + AD² -  ED²            

AB² = AD² + (BE- ED)  (BE+ ED )

AB² = AD² + (BE- ED)   BD             (3)

In        AEC :-

AC² = EC² + AE² 

AC² = EC² + AD² -  ED²            (From eqn. 2)

AC² = AD² + (EC- ED)  (EC+ ED )

AC² = AD² + DC(EC+ED)

Now DC=BD as D is mid point of BC

AC² = AD² + BD(EC+ED)                  (4)

Now Adding eqn. 3 & 4

AB² + AC²= AD² + (BE- ED)   BD   + AD² + BD(EC+ED)

AB² + AC²= 2 AD² + BD(BE-ED+EC+ED)

AB² + AC²= 2 AD² + BD ( BE+EC)

AB² + AC²= 2 AD² + BD 2BD

AB² + AC²= 2 AD² + 2 BD²        (H.P)