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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, 4 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, 4 months ago

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@Jon Haussmann Can you please provide the proof sir ? Karthik Venkata · 2 years, 4 months ago

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@Jon Haussmann 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, 4 months ago

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@Pranav Kirsur In a quadrilateral, the conditions "one diagonal bisects the area" and "one diagonal bisects the other diagonal" are equivalent. Jon Haussmann · 2 years, 4 months ago

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@Jon Haussmann Oh yes, thank you for pointing that out to me Pranav Kirsur · 2 years, 4 months ago

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@Pranav Kirsur Yeah , thats why I posted this as a note. Nihar Mahajan · 2 years, 4 months ago

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@Nihar Mahajan Did you get a proof Pranav Kirsur · 2 years, 4 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, 4 months ago

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@Nihar Mahajan yes,same Pranav Kirsur · 2 years, 4 months ago

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@Jon Haussmann Sir, thanks for your answer.But it may be more beneficial for us if you post a nice proof. Nihar Mahajan · 2 years, 4 months ago

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@Nihar Mahajan Now that you know the right condition, you should try proving it. Jon Haussmann · 2 years, 4 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, 4 months ago

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@Nihar Mahajan Nihar, Tujhe pata hai bhai ki Mujhe Geometry nahi aati :3 Jo Bhi Ho. Thanks For @Mentioning Me :D Mehul Arora · 2 years, 4 months ago

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@Mehul Arora I mentioned you so that you will have something interesting in geometry. ;) Nihar Mahajan · 2 years, 4 months ago

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@Nihar Mahajan Yeah, Thanks! :) :D :) Mehul Arora · 2 years, 4 months ago

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