The area of a triangle is given by the formula ...
... A = (1/2)bh
where b is the base length and h is the height perpendicular to the base.
If we let the length of the sides of the square be represented by s, then we have ...
... AM = MB = MO = s/2
and the areas of the triangles are ...
... AAMD = (1/2)(s)(s/2) = s²/4
... ABMC = (1/2)(s)(s/2) = s²/4
... ACDO = (1/2)(s)(s/2) = s²/4
and the area of the quadrilateral is the sum of the areas of triangles MOC and MOD, so is ...
... AMCOD = (1/2)(s/2)(s/2) +(1/2)(s/2)(s/2) = s²/8 +s²/8 = s²/4
Hence the described areas are equal to each other. (All are s²/4.)
