# A geometric inequality

In a $$\Delta$$ABC if $$P$$ is any interior point inside it, Find the minimum value of :

$$\displaystyle \text{min}(BC.PA^2+CA.PB^2+AB.PC^2)$$

1 year, 7 months ago

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keep 3 masses of values BC,CA,AB at points A,B,C. Moment of inertia about any point P is given by (BC.PA^2+...)which is what we have to minimize. However moment of inertia is minimum at centre of mass so we find centre of mass whose coordinates are Same as formula of coordinates of incentre given sides. So the required point P is its incentre

- 1 year, 7 months ago

Yep, That's a great observation.

- 1 year, 7 months ago

What was ur solution?

- 1 year, 7 months ago

That used the Erdos-Mordell inequality and then some co-ordinate, anyways his observation from physics outweighs mine here.

- 1 year, 7 months ago

Awesome solution!

- 1 year, 7 months ago

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