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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 $△HCE$, $$∠AHB=∠CHE…
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Souzume Hachimangu Shrine (1861), Souzume, Northern Ward, Okayama City, Okayama Prefecture (05)
Problem Find the length of one side of a cube that is $1881676371789154860897069 \ cubic \ inches$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution When $1881676371789154860897069$ is decomposed into prime factors, $$3^6×3607^3×3803^3=(3^2×3607×3803)^3.…
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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…