Question

The midpoints of the sides of two convex quadrangles coincide. Prove that the areas of the quadrangles are equal.

Answer 1

The polygon $AEQFCGSH$ is common area of both the quadrangles.

We see that $△BFQ\equiv △CFR\text{as}BF=FC,QF=FR,\angle BFQ=\angle CFR$. Thus areas of $△BFQ$ and $△CFR$ are equal.

We see that $△BEQ\equiv △AEP\text{as}BE=AE,PE=EQ,\angle BEQ=\angle AEP$. Thus areas of $△BEQ$ and $△AEP$ are equal.

We see that $△AHP\equiv △SHD\text{as}AH=HD,PH=HS,\angle AHP=\angle SHD$.Thus areas of $△AHP$ and $△SHD$ are equal.

We see that $△SGD\equiv △CGR\text{as}SG=GR,DG=GC,\angle SGD=\angle CGR$. Thus areas of $△SGD$ and $△CGR$ are equal.

area of $\square ABCD=AEQFCGSH+△EBQ+△BFQ+△SGD+△SHD$

$=AEQFCGSH+△EAP+△CFR+△CGR+△AHP=\square PQRS$