Answer :


Given: AB = BC = CA = x


ADC = ADB = 90°


To prove: AB2 + BC2 + CA2 = 4AD2


Proof:


We know that in an equilateral triangle perpendicular from any vertex on opposite side bisects it.


So, BD = DC = 1/2BC ……………… (1)


By applying Pythagoras theorem in ΔABD, we get,


AB2 = AD2 + BD2 [ H2 = P2 + B2]


By substituting BD from eqn. (1) we get,


AB2 = AD2 + (1/2BC)2


AB2 = AD2 + 1/4BC2


4AB2 = 4AD2 + BC2


4AB2 - BC2 = 4AD2


4AB2 - AB2 = 4AD2 [Given: AB = BC]


3AB2 = 4AD2


AB2 + BC2 + CA2 = 4AD2 [ AB = BC = CA]


Hence proved.


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