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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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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Souzume Hachimangu Shrine (1861), Souzume, Northern Ward, Okayama City, Okayama Prefecture (04)
Problem What is the length of one side of a square inscribed in a triangle, of which the longest side is $21 \ inches$ and the perpendicular line from the vertex to the longest side is $8 \ inches$? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$…
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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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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Souzume Hachimangu Shrine (1861), Souzume, Northern Ward, Okayama City, Okayama Prefecture (03)
Problem Find the diameter of a circle inscribed in a triangle, of which sides have lengths of $21, \ 17$ and $10 \ inches$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let…
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The Encyclopedia of Geometry (0196)
Problem In a parallelogram,$(1)$ opposite angles are equal to each other;$(2)$ opposite sides are equal in length. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution $(1)$ If we draw the diagonal $AC$ on the parallelogram $ABCD$, since $AB∥DC$,$$∠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 Kibi 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, \…