Answer :

We know, By Apollonius theorem

In ΔABC**,** if L is the midpoint of side AC, then AB^{2} + BC^{2} = 2BL^{2} + 2AL ^{2}

Given that, BL is median i.e. L is the mid-point of CA

⇒ AB^{2} + BC^{2} = 2BL^{2} + 2AL^{2}

[1]

Also, if M is the midpoint of side AB, then AC^{2} + BC^{2} =2CM^{2} + 2BM^{2}

Given that, CM is median i.e. M is the mid-point of BA

⇒ AC^{2} + BC^{2} = 2CM^{2} + 2BM^{2}

[2]

Also, In ΔABC, By Pythagoras theorem i.e.

(Hypotenuse)^{2} = (base)^{2} + (Perpendicular)^{2}

⇒ BC^{2} = AC^{2} + AB^{2} [3]

Adding [1] and [2]

⇒ AB^{2} + AC^{2} + 4BC^{2} = 4(BL^{2} + CM^{2})

⇒ BC^{2} + 4BC^{2} = 4(BL^{2}+ CM^{2}) [From 3]

⇒ 5BC^{2} = 4(BL^{2}+ CM^{2})

Hence Proved.

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