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Hioki Shrine (1911), Shinshu-shinmachi, Nagano City, Nagano Prefecture (04)
Problem You purchased $400$ pieces of lumber for $40$ dollars. The pieces were high-quality at $0.16$ dollars each and low-quality at $0.08$ dollars each. How many pieces of high-quality lumber did you buy at this time? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0212)
Problem Within a parallelogram $ABCD$, draw $OA, \ OB, \ OC$, and $OD$ by connecting any point $O$ to each of the vertices, and let the midpoints of each be $E, \ F, \ G$, and $H$. Then, if the intersections of $DE$ and $CF$, $AF$ and $DG$, $AH$ and $BG$, and $BE$ and $CH$…
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Hioki Shrine (1911), Shinshu-shinmachi, Nagano City, Nagano Prefecture (03)
Problem There are three squares: large, medium, and small. The sum of the areas of the three squares is $133 \ in^2$. The medium square is $20 \ in^2$ larger than the small square, and the large square is $45 \ in^2$ larger than the medium one.Find the length of one side of the small…
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Aoki Hachiman Shrine (1854), Nishinoura, Tsurajima-cho, Kurashiki City, Okayama Prefecture (01)
According to Yamakawa, the location of Aoki Hachiman Shrine is Aoki, Nishinoura, Tsurajima-cho, Kurashiki City, Okayama Prefecture, but this cannot be confirmed. Problem As shown in the figure, there are two large circles and one small circle with their centers on a line, and the small circle is between the square and the large circles.If…
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The Encyclopedia of Geometry (0211)
Problem If equilateral triangles $ABD$ and $ACE$ are drawn outside a triangle $ABC$, and an equilateral triangle $BCF$ with the side $BC$ is drawn on the same side as a vertex $A$, then the quadrilateral $AEFD$ is a parallelogram. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0209)
Problem When you draw four squares with each side as one side on the outside of the parallelogram $ABCD$ and connect the centers of these squares to form a quadrilateral, it will be a square. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0210)
Problem If you draw the equilateral triangles $ABE$ and $CDF$ on the outside of a parallelogram $ABCD$ and the equilateral triangle $BCG$ on the same side as the parallelogram, then $$EG=AC \qquad and \qquad FG=DB.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0208)
Problem The sum of the lengths of the perpendicular lines drawn from the two vertices $A$ and $C$ of a parallelogram $ABCD$ to the line $XY$ outside the quadrilateral is equal to the sum of the lengths of the perpendicular lines drawn from the other two vertices $B$ and $D$ to $XY$. $$ $$ $$ $$ $\downarrow$…
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Hioki Shrine (1911), Shinshu-shinmachi, Nagano City, Nagano Prefecture (02)
Problem You borrowed the principal of $15000\ yen$ with annual compound interest, and were able to pay off the loan by repaying $6000 \ yen$ after one year, $8400 \ yen$ after two years, and $7200 \ yen$ after three years. Find the annual interest rate in this case. $$ $$ $$ $$ $\downarrow$ $\downarrow$…
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The Encyclopedia of Geometry (0207)
Problem For the parallelogram $ABCD$, in which $2AB=AD$, take points $E$ and $F$ such that $AE=AB=BF$ by extending $AB$, and let $G$ be the intersection point of $EC$ and $FD$. Then, $$∠EGF=∠R.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…