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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 $AB∥DC$,$$∠CAB=∠ACD \qquad…
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The Encyclopedia of Geometry (0195)
Problem The lines joining the midpoints of two pairs of opposite sides of a quadrilateral intersect at the midpoint of the line segment joining the midpoints of the two diagonals. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let…
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Souzume Hachimangu Shrine (1861), Souzume, Northern Ward, Okayama City, Okayama Prefecture (02)
Problem What is the side length of a square with an area of $85000 ft^2$ ? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution The prime factorization of $85000$ is:$$85000=2^3×5^4×17=2^2×5^4×(2×17),$$$$∴ \ \sqrt{85000}=2×5^2×\sqrt{2×17}≒291.547594742265.$$ …
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Souzume Hachimangu Shrine (1861), Souzume, Northern Ward, Okayama City, Okayama Prefecture (01)
Souzume Hachimangu Shrine is located 1.2 $km$ west-southwest of Kibitsu Station on the JR Kibitsu Line. Problem The contents are unknown as some parts are illegible. Reference Yoshikazu Yamakawa, ed. (1997) Okayama ken no Sangaku (Sangaku in Okayama Prefecture), p.27; p.400.
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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…
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The Encyclopedia of Geometry (0192)
Problem In a quadrilateral $ABCD$, if the angles of intersection of the bisectors of $∠A$ and $∠B$, and $∠A$ and $∠C$ are $α$ and $β$, respectively, then $$α=\frac{1}{2} (∠C+∠D) \qquad and \qquad β=\frac{1}{2} |∠B-∠D|.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0191)
Problem In a quadrilateral $ABCD$ such that $AB=CD$, if the midpoints of $DA$ and $BC$ are $P$ and $Q$ respectively, and the midpoints of the diagonals $AC$ and $BD$ are $M$ and $N$ respectively, then $PQ$ and $MN$ are perpendicular to each other. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0190)
Problem The length of the line segment connecting the midpoints $P$ and $Q$ of the opposing sides $AB$ and $CD$ of a quadrilateral $ABCD$ is not greater than half the sum of the lengths of the other two sides $BC$ and $DA$. If this line segment $PQ$ is equal to the sum of the lengths…
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The Encyclopedia of Geometry (0189)
Problem If the lengths of the two sides $AB$ and $CD$ of a quadrilateral $ABCD$ are equal, then the extensions of these sides and the line connecting the midpoints $M$ and $N$ of the other two sides $AD$ and $BC$ will make equal angles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$…