Problem For the parallelogram $ABCD$, in which $2AB=AD$, take points $E$ and $F$ such that $AE=AB=BF$ by extending $AB$, and let $G$ be the intersection point of $EC$ and $FD$. Then, $$∠EGF=∠R.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Let $E$ and $F$ be the midpoints of sides $AB$ and $BC$ of a triangle $ABC$, respectively. Put points $G$ and $H$ on $AC$ so that $AG=GH=HC$, and let $D$ be the intersection point of lines $EG$ and $FH$. Then, the quadrilateral $ABCD$ is a parallelogram. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$…

Problem If the perpendicular lines drawn from the vertices $A, \ B, \ C$ and $D$ of the parallelogram $ABCD$ to the diagonal $AC$ or $BD$ have feet $F, \ E, \ H$ and $G$ respectively, then the quadrilateral $EFGH$ is a parallelogram. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem The lengths of the portions cut by the two straight lines passing through the intersection $O$ of the diagonals of the parallelogram $ABCD$ from the two opposite sides $AD$ and $BC$ or their extensions are equal. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If points $E$ and $F$ are placed on the diagonal $AC$ of a parallelogram $ABCD$ such that $AE = CF$, then $BEDF$ is also a parallelogram. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△AED$ and $△CFB$,…

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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…