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!

## Comments

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TopNewestSuch 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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– Karthik Venkata · 2 years, 4 months ago

Can you please provide the proof sir ?Log in to reply

– Pranav Kirsur · 2 years, 4 months ago

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.Log in to reply

– Jon Haussmann · 2 years, 4 months ago

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

– Pranav Kirsur · 2 years, 4 months ago

Oh yes, thank you for pointing that out to meLog in to reply

– Nihar Mahajan · 2 years, 4 months ago

Yeah , thats why I posted this as a note.Log in to reply

– Pranav Kirsur · 2 years, 4 months ago

Did you get a proofLog in to reply

– Nihar Mahajan · 2 years, 4 months ago

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.Log in to reply

– Pranav Kirsur · 2 years, 4 months ago

yes,sameLog in to reply

– Nihar Mahajan · 2 years, 4 months ago

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

– Jon Haussmann · 2 years, 4 months ago

Now that you know the right condition, you should try proving it.Log in to reply

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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@Calvin Lin @Brian Charlesworth @Trevor Arashiro@Pi Han Goh @Chew-Seong Cheong @Mehul Arora @Sharky Kesa @Sanjeet Raria @Sudeep Salgia @Ronak Agarwal @everyone help us. – Nihar Mahajan · 2 years, 4 months ago

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– Mehul Arora · 2 years, 4 months ago

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

– Nihar Mahajan · 2 years, 4 months ago

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

– Mehul Arora · 2 years, 4 months ago

Yeah, Thanks! :) :D :)Log in to reply