Divide convex polygon into 4 equal areas

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Homework Statement



Show that it is possible to cut any convex polygon into 4 pieces of equal areas by using two cuts perpendicular to each other.

Homework Equations


None, it's just a proof I found on the back of my book. The relevant chapter is Continuity, the maximum principle, and intermediate value theorem for real analysis.


The Attempt at a Solution



I have no idea! Can someone tell me how to approach this please?

Thanks!
 
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Anyone know how to do this?
 
What about considering the centroid of the convex polygon?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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