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Katayama-hiko Shrine (1873), Osafune-cho, Setouchi City, Okayama Prefecture (01)
Katayama-hiko Shrine is located $850$ meters southeast from Osafune Station on the JR Ako Line. Problem As shown in the figure, two mutually circumscribed circles of diameter $d$ are inscribed in a fan of radius $R$. $c$ is the common tangent of these two circles, and is also the chord of the fan that inscribes…
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The Encyclopedia of Geometry (0033)
Problem Two triangles with equal bases, vertex angles, and heights are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△ABC$ and $△A’B’C’$, suppose that $$BC=B’C’, \quad AD=A’D’ \quad and \quad ∠BAC=∠B’A’C’.$$…
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The Encyclopedia of Geometry (0032)
Problem Two right triangles which have equal hypotenuses and the other equal sides respectively are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Suppose that $AC=DF$ and $AB=DE$ in $⊿ABC$ and $⊿DEF$…
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The Encyclopedia of Geometry (0031)
Problem Two right triangles which have equal hypotenuses and equal acute angles are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $∠B=∠E=∠R$, $AC=DF$ and $∠C=∠F$. Place $D$ on top of $A$,…
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The Encyclopedia of Geometry (0030)
Problem Two triangles, in which the three corresponding sides are equal respectively, are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution In $△ABC$ and $△DEF$, $$AB=DE, \quad BC=EF \quad and \quad AC=DF.$$…