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The Encyclopedia of Geometry (0202)
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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0201)
Problem If the midpoints of sides $CD$ and $AD$ of a parallelogram $ABCD$ are $E$ and $F$ respectively, then $BE$ and $BF$ trisect the diagonal $AC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△HAB$ and $△HCE$, $$∠AHB=∠CHE…
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The Encyclopedia of Geometry (0200)
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 $AB=DC$…
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The Encyclopedia of Geometry (0199)
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…
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The Encyclopedia of Geometry (0198)
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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0197)
Problem A necessary and sufficient condition for a quadrilateral $ABCD$ to be a parallelogram is that for any point $P$ in the quadrilateral, the following holds: $$△PAB+△PCD=\frac{1}{2}◻ABCD. \qquad [*]$$ Show that this condition is necessary and sufficient. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0196)
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 $AB∥DC$,$$∠CAB=∠ACD \qquad…
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The Encyclopedia of Geometry (0195)
Problem The lines joining the midpoints of two pairs of opposite sides of a quadrilateral intersect at the midpoint of the line segment joining the midpoints of the two diagonals. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let…
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The Encyclopedia of Geometry (0194)
Problem The opposite angles of the quadrilateral formed by the four straight lines that bisect the exterior angles of a quadrilateral are supplementary to each other. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $E, \ F, \…
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The Encyclopedia of Geometry (0193)
Problem When the bisectors of each angle of a quadrilateral intersect to form a second quadrilateral, $(1)$ the sum of the two opposing angles of the quadrilateral is two right angles; $(2)$ if the original quadrilateral is a parallelogram, the second quadrilateral is a rectangle, and its two diagonals are parallel to each side of…