Characterization of Trapezoid
What Might This Be About?
The midline $EF$ of a convex quadrilateral $ABCD$ divides its area into equal parts.
Prove that $ABCD$ is a trapezoid.
Let $[X]$ denote the area of figure $X.$
Since $AE=BE,$ $[\Delta AEF]=[\Delta BEF].$ Since it is given that $[ACFE]=[BDFE],$ taking the difference we see that $[\Delta ACF]=[\Delta BDF].$
But the two triangles have equal bases $CF=DF,$ implying that their altitudes to the bases are also equal, i.e., $A$ and $B$ are at the sam distance from $CD.$ Hence $AB\parallel CD.$
V.V. Proizvolov, Zadachi na vyrost (Russian. Problems to grow up with), Miros, Moscow, 1995, ISBN 5708401133, p 7
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