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Question

If O is any point within ABC,then to prove that AB + BC + CA > AO + BO + CO

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Solution

Let us ABC and mark point O anywhere inside the triangle.

Now, there is a known inequality:
AO + CO < AB + BC (Proof can be given, if you need)
Similarly,
AO + BO < AC + BC
BO + CO > AB + AC
Adding all these together, we get,
2(AO + BO + CO) < 2 (AB + BC + CA)
AB+BC+CA>AO+BO+CO

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