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Find sufficient and necessary conditions on the sides and angles of a convex $n$-gon $A_1A_2 \cdots A_n$ ,so that there is an inner point $M$ such that two perpendicular lines through $M$ divide the $n-$gon $A_1A_2 \cdots A_n$ into four equal polygons of equal area . Determine point $M$ .

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Do you mean the four component polygons to be congruent? –  Mark Bennet Jun 18 at 9:24
    
What's the source of this problem, please? –  Gerry Myerson Jun 18 at 9:44
    
WHAT'S THE SOURCE OF THIS PROBLEM, PLEASE? –  Gerry Myerson Jun 19 at 13:16
    
@GerryMyerson , it was from some textbook I saw in a library , whose name I have forgotten –  Shivam Patel Jun 19 at 16:21
    
Is this a duplicate of math.stackexchange.com/questions/601228/… ? –  Gerry Myerson Jun 19 at 23:32

1 Answer 1

up vote 2 down vote accepted

If all you want is equal area, then such a dissection always exists.

Claim: given any angle $\theta$, there always exists a unique line $l_\theta$ with that slope such that the area of the figure is equally split.

Claim: as $l_\theta$ varies continuously wrt theta.

Now, consider $l_\theta, l_{\theta+90^\circ}$. It divides the figure into 4 parts where the opposite areas are the same.

Apply a continuity argument + intermediate value theorem to show that there is an angle where all 4 parts have equal area.

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