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The Encyclopedia of Geometry (0219)
Problem The diagonal of a rectangle is longer than any line segment drawn between its opposite sides. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $EF$ be any line segment drawn between $AB$ and $CD$. If we draw…
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The Encyclopedia of Geometry (0218)
Problem If a parallelogram whose sides are parallel to the diagonals of a rectangle $ABCD$ is inscribed in the rectangle, the perimeter of the parallelogram will always be constant. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $K$…
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Hioki Shrine (1911), Shinshu-shinmachi, Nagano City, Nagano Prefecture (07)
Problem As shown in the figure, a large circle and a small circle that circumscribe each other are inscribed in a square. If the area of the square is $235 \ in^2$ and the difference between the diameters of the large and small circles is $2 \ in$, find the diameter of the small circle.…
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The Encyclopedia of Geometry (0217)
Problem A quadrilateral formed by connecting the points where two straight lines that pass through the intersection of the diagonals of a parallelogram and intersect with each other at right angles intersect with each side is a rhombus. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0216)
Problem If the intersection of the bisectors of the two vertex angles $B$ and $C$ of the rectangle $ABCD$ is $E$, and similarly, the intersections of the bisectors of $A$ and $B$, $D$ and $A$, and $C$ and $D$ are $F, \ G$, and $H$, respectively, then the quadrilateral $FEHG$ is a square. $$ $$ $$ $$…
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Hioki Shrine (1911), Shinshu-shinmachi, Nagano City, Nagano Prefecture (06)
Problem You want to buy equal amounts of high-grade, medium-grade, and low-grade rice for 5 yen, but for every 1 yen you spend, you get 7 liters of high-grade rice, 8 liters of medium-grade rice, and 10 liters of low-grade rice, respectively. In this case, how many liters of each grade of rice can you…
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The Encyclopedia of Geometry (0215)
Problem A convex quadrilateral $ABCD$ is a rhombus if $∠A=∠C, \ ∠B=∠D$ and $AB=BC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution The sum of the interior angles of a quadrilateral is $360°$. Thus, if $∠A = ∠C$ and…
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Hioki Shrine (1911), Shinshu-shinmachi, Nagano City, Nagano Prefecture (05)
Problem As shown in the figure, the square $BDEF$ is inscribed in the right triangle $ABC$, and the sum of the areas of $⊿ADE$ and $⊿EFC$ is $150 \ in^2$, and the length of the side $AB$ is $21 \ in$. Find the length of one side of the square $BDEF$. $$ $$ $$ $$…
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The Encyclopedia of Geometry (0214)
Problem The two diagonals of a rhombus are perpendicular to each other and bisect each vertex angle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution $△ABD$ and $△CDB$ share $BD \ (=DB)$, and $$AB=CD \qquad and \qquad AD=CB,$$ $$∴ \quad…
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The Encyclopedia of Geometry (0213)
Problem The two diagonals of a rectangle are equal. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△ABC$ and $△DCB$,$$AB=DC, \qquad BC=CB \qquad and \qquad ∠B=∠C \ (=∠R),$$$$∴ \quad △ABC≡△DCB,$$$$∴ \quad AC=DB.$$ $ $ $ $ $…