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The Encyclopedia of Geometry (0237)
Problem If $M$ is the midpoint of side $AB$ of right triangle $ABC \ (∠C=∠R)$, and the squares $BCDE$ and $ACFG$ are constructed externally on sides $BC$ and $CA$, respectively, then $$CM=\frac{1}{2} DF \qquad and \qquad CM⊥DF.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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Bitchu-no-kuni Soja-gu Shrine (1853), Soja, Soja City, Okayama Prefecture (01)
Bitchu-no-kuni Soja-gu Shrine is located $550 \ m$ southeast of Higashi-Soja Station on the JR Kibi Line. Problem As shown in the figure, a rectangle contains two large circles and two small circles separated by diagonals.If the diameter of the large circle is $10 \ cm$ and that of the small circle is $7.2…
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The Encyclopedia of Geometry (0236)
Problem Construct squares $ABDE$ and $ACFG$ externally on triangle $ABC$, with sides $AB$ and $AC$ as their respective bases. Let $H$ be the foot of the perpendicular from $A$ to $BC$. Let $M$ be the point where line $HA$ intersects segment $EG$. Then, show that $$EM = MG.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…
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The Encyclopedia of Geometry (0235)
Problem If squares $ABDE$ and $ACFG$ are constructed externally on sides $AB$ and $AC$ of triangle $ABC$, then the areas of $△ABC$ and $△AEG$ are equal. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let the areas of…
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Tsukama Shrine (1873), Tsukama, Matsumoto City, Nagano Prefecture (05)
Problem As shown in the figure, inside the right triangle $ABC$ (with $∠B = ∠R$), there is another right triangle that shares the shorter side $AB$. In addition, two congruent circles are placed on the left and right sides of triangle $ABC$, each tangent to the two line segments drawn from vertices $A$ and $B$…
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The Encyclopedia of Geometry (0234)
Problem Take a point $E$ on the line through vertex $C$ of square $ABCD$ that is parallel to $BD$, and assume that $BE=BD$. Let $F$ be the intersection of $BE$ with $CD$. Then $$DE=DF.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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Tsukama Shrine (1873), Tsukama, Matsumoto City, Nagano Prefecture (04)
Problem As shown in the figure, the upper and lower congruent circles are mutually tangent, and the lower circle is tangent to a line.The left and right congruent circles are each tangent to both the upper and lower circles.A small circle is tangent to the two congruent circles and also tangent to a line.If the…
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The Encyclopedia of Geometry (0233)
Problem Let $E$ be a point on diagonal $BD$ of square $ABCD$ such that $BE=BC$. From any point $P$ on segment $CE$, drop perpendiculars to $BD$ and $BC$, and let $F$ and $G$ be the respective feet. Then $$PF+PG=\frac{1}{2} BD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…