Question

prove that the area of the figure obtained by joining the mid points of the sides of a quadrilateral is half than that of the quadrilateral

Answer 1

EFEF is the basis media of triangle ABCABC, and then A(BEF)=1/4A(ABC)A(BEF)=1/4A(ABC). Idem A(GHD)=1/4A(ACD)A(GHD)=1/4A(ACD).

Now, note that A(ACD)+A(ABC)=A(ABCD)→A(BEF)+A(GHD)=1/4A(ABCD)[∗1]A(ACD)+A(ABC)=A(ABCD)→A(BEF)+A(GHD)=1/4A(ABCD)[∗1].

Idem, A(HEA)+A(GFC)=1/4A(ABCD)[∗2]A(HEA)+A(GFC)=1/4A(ABCD)[∗2].

Adding [∗1]+[∗2][∗1]+[∗2] the problem is solved