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Why are we caring so much about the existence of such a tiny Point P?

Consider a quadrilateral \(ABCD\) . Find the necessary and sufficient condition with proof so that there exists a point \(P\) in the interior of \(ABCD\) such that \(A(PAB)=A(PBC)= A(PCD)= A(PDA)\).

\(\text{A( ) represents Area}\)

Nice solutions are always welcome!

Note by Nihar Mahajan
2 years, 7 months ago

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Such a point \(P\) exists if and only if one diagonal bisects the area of the quadrilateral.

Jon Haussmann - 2 years, 7 months ago

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Can you please provide the proof sir ?

Karthik Venkata - 2 years, 7 months ago

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Sir, actually this problem is from the homework of our class, and our sir has a different condition- P exists iff one diagonal bisects the other.

Pranav Kirsur - 2 years, 7 months ago

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In a quadrilateral, the conditions "one diagonal bisects the area" and "one diagonal bisects the other diagonal" are equivalent.

Jon Haussmann - 2 years, 7 months ago

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@Jon Haussmann Oh yes, thank you for pointing that out to me

Pranav Kirsur - 2 years, 6 months ago

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Yeah , thats why I posted this as a note.

Nihar Mahajan - 2 years, 7 months ago

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@Nihar Mahajan Did you get a proof

Pranav Kirsur - 2 years, 7 months ago

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@Pranav Kirsur I got the proof for the condition- P exists iff one diagonal bisects the other.I did not get proof for the condition that Jon Hausmann has posted.

Nihar Mahajan - 2 years, 7 months ago

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@Nihar Mahajan yes,same

Pranav Kirsur - 2 years, 7 months ago

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Sir, thanks for your answer.But it may be more beneficial for us if you post a nice proof.

Nihar Mahajan - 2 years, 7 months ago

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Now that you know the right condition, you should try proving it.

Jon Haussmann - 2 years, 7 months ago

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me , @Kalash Verma @Harsh Shrivastava @CH Nikhil are in need of a nice solution.Thanks!

Nihar Mahajan - 2 years, 7 months ago

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Nihar, Tujhe pata hai bhai ki Mujhe Geometry nahi aati :3 Jo Bhi Ho. Thanks For @Mentioning Me :D

Mehul Arora - 2 years, 7 months ago

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I mentioned you so that you will have something interesting in geometry. ;)

Nihar Mahajan - 2 years, 7 months ago

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@Nihar Mahajan Yeah, Thanks! :) :D :)

Mehul Arora - 2 years, 7 months ago

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