Problem If the midpoints of opposite sides $AB$ and $CD$ of a parallelogram $ABCD$ are $E$ and $F$ respectively, then the quadrilateral formed by the four straight lines connecting these two points and both ends of the opposite sides is also a parallelogram. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If the midpoints of sides $CD$ and $AD$ of a parallelogram $ABCD$ are $E$ and $F$ respectively, then $BE$ and $BF$ trisect the diagonal $AC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△HAB$ and $△HCE$, $$∠AHB=∠CHE…

Problem If the midpoints of opposite sides $AB$ and $DC$ of a parallelogram $ABCD$ are $E$ and $F$ respectively, then $DE$ and $BF$ trisect the diagonal $AC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Since $AB∥DC$,$$EB∥DF.$$Also, since $AB=DC$…

Problem If each vertex of a parallelogram $PQRS$ is on each side of another parallelogram $ABCD$, then the diagonals of the two parallelograms pass through the same points. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△APS$ and…

Problem If points $P, \ Q, \ R$ and $S$ are taken on the sides $AB, \ BC, \ CD$ and $DA$ of a parallelogram $ABCD$ such that $AP=BQ=CR=DS$, then the quadrilateral $PQRS$ is also a parallelogram. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem A necessary and sufficient condition for a quadrilateral $ABCD$ to be a parallelogram is that for any point $P$ in the quadrilateral, the following holds: $$△PAB+△PCD=\frac{1}{2}◻ABCD. \qquad [*]$$ Show that this condition is necessary and sufficient. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem In a parallelogram,$(1)$ opposite angles are equal to each other;$(2)$ opposite sides are equal in length. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution $(1)$ If we draw the diagonal $AC$ on the parallelogram $ABCD$, since $AB∥DC$,$$∠CAB=∠ACD \qquad…