TITLE: How to prove a+c=b+d
QUESTION [1 upvotes]: Let $ABCD$ be a quadrilateral. Let $A′$ be the midpoint of $AB$, $B′$ the mid-point of $BC$, $C′$ the mid-point of $CD$ and $D′$ the mid-point of $AD$. Draw the lines $A′C′$ and $B′D′$ and let $a, b, c, d$ be the areas of the four minor quadrilaterals as shown in the figure. Prove that $a + c = b + d$.
I’ve realised that $a$ and $b$ have one of the same bases, but the other base is different. I think it has something to with $b$ sharing a base with $c$, and vice versa with $a$ and $d$.

REPLY [1 votes]: Hint:
In this figure there are some triangles of same color that have the same area.
 Can you see why?
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