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!

#Geometry

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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

Jon Haussmann - 6 years, 1 month ago

Can you please provide the proof sir ?

Venkata Karthik Bandaru - 6 years, 1 month ago

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

Nihar Mahajan - 6 years, 1 month ago

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

Jon Haussmann - 6 years, 1 month 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.

Pranav Kirsur - 6 years ago

Yeah , thats why I posted this as a note.

Nihar Mahajan - 6 years ago

@Nihar Mahajan – Did you get a proof

Pranav Kirsur - 6 years ago

@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 - 6 years ago

@Nihar Mahajan – yes,same

Pranav Kirsur - 6 years ago

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

Jon Haussmann - 6 years ago

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

Pranav Kirsur - 6 years ago

@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 - 6 years, 1 month ago

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

Mehul Arora - 6 years, 1 month ago

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

Nihar Mahajan - 6 years, 1 month ago

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

Mehul Arora - 6 years, 1 month ago

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

Nihar Mahajan - 6 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$