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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 $$PF+PG=\frac{1}{2} BD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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Tsukama Shrine (1873), Tsukama, Matsumoto City, Nagano Prefecture (03)
Problem As shown in the figure, two congruent circles are inscribed in an isosceles trapezoid. If the upper base of the trapezoid is $2.4 \ inches$ and the lower base is $12 \ inches$, find the diameters of the circles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0232)
Problem Let rectangle $PQRS$ be inscribed in square $ABCD$. Each side of the rectangle is parallel to a diagonal of the square. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $O$ be the intersection point of the…
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The Encyclopedia of Geometry (0231)
Problem Draw squares $ABEF$ and $ACGH$ externally on triangle $ABC$. On the same side of the triangle, construct an isosceles right triangle $BCP$ with $BC$ as its hypotenuse. Then points $E, \ P$ and $G$ are collinear, and $$EP=PG.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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Aoki Hachiman Shrine (1854), Nishinoura, Tsurajima-cho, Kurashiki City, Okayama Prefecture (02)
Problem As shown in the figure, an isosceles triangle, two large circles, and one small circle are inscribed in a square.If one side of the square measures $10 \ inches$, and the difference between the lengths of the equal sides and the base of the isosceles triangle is $5 \ inches$, find the lengths of…
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The Encyclopedia of Geometry (0230)
Problem Draw squares $ABDM$ and $ACEN$ externally on sides $AB$ and $AC$ of triangle $ABC$, respectively. If perpendiculars $DF$ and $EG$ are dropped from vertices $D$ and $E$ to line $BC$, then the length of $BC$ equals the sum of $DF$ and $EG$, and the area of triangle $ABC$ equals the sum of the areas…