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The Encyclopedia of Geometry (0208)
Problem The sum of the lengths of the perpendicular lines drawn from the two vertices $A$ and $C$ of a parallelogram $ABCD$ to the line $XY$ outside the quadrilateral is equal to the sum of the lengths of the perpendicular lines drawn from the other two vertices $B$ and $D$ to $XY$. $$ $$ $$ $$ $\downarrow$…
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Hioki Shrine (1911), Shinshu-shinmachi, Nagano City, Nagano Prefecture (02)
Problem You borrowed the principal of $15000\ yen$ with annual compound interest, and were able to pay off the loan by repaying $6000 \ yen$ after one year, $8400 \ yen$ after two years, and $7200 \ yen$ after three years. Find the annual interest rate in this case. $$ $$ $$ $$ $\downarrow$ $\downarrow$…
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The Encyclopedia of Geometry (0207)
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, $$∠EGF=∠R.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0206)
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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$…
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The Encyclopedia of Geometry (0205)
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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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Hioki Shrine (1911), Shinshu-shinmachi, Nagano City, Nagano Prefecture (01)
Hioki Shrine is located $23 \ km$ west of JR Shinonoi Station. Problem As shown in the figure, a circle is inscribed in an equilateral triangle. If the square of one side of the equilateral triangle is $64 \ in^2$, find the diameter of the inscribed circle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$…
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The Encyclopedia of Geometry (0204)
Problem The lengths of the portions cut by the two straight lines passing through the intersection $O$ of the diagonals of the parallelogram $ABCD$ from the two opposite sides $AD$ and $BC$ or their extensions are equal. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0203)
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 $△CFB$,…
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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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$…