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The Encyclopedia of Geometry (0038)
Problem When two sides of a triangle are unequal, the angle opposite the longer side is greater than the angle opposite the shorter side. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Suppose…
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The Encyclopedia of Geometry (0037)
Problem Regarding two congruent triangles $△ABC$ and $△A’B’C’$, when the two pairs of sides $AB$ and $A’B’$ and $AC$ and $A’C’$ are perpendicular to each other, the remaining pair $BC$ and $B’C’$ are also perpendicular to each other. However, assume that $∠A$ and $∠A’$ are not right angles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$…
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The Encyclopedia of Geometry (0036)
Problem There are $△ABC$ and $△A’B’C’$. $$AB=A’B’, \ BC=B’C’, \ and \ ∠A=∠A’.$$ Then, under which of the following conditions can we always have $△ABC≡△A’B’C’$? (1) Either $△ABC$ or $△A’B’C’$ is an obtuse triangle. (2) $∠C$ is a right angle. (3) $BC>AB$. (4) $AC$ and $A’C’$ are the maximum sides of $△ABC$ and $△A’B’C’$, respectively. (5) $AB>AC$…
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Katayama-hiko Shrine (1873), Osafune-cho, Setouchi City, Okayama Prefecture (02)
Problem There is a right triangular piece of land with a short side of $30\ m$ and a long side of $40 \ m$. As shown in the diagram, add $2 \ m$ wide roads to this, and divide the remaining area equally into three sections, $S_1, S_2$, and $S_3$. What are their shapes? And…
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The Encyclopedia of Geometry (0035)
Problem When the two sides of one of two triangles are equal to the two sides of the other, and the angles to one set of equal sides are also equal, the angles to the other set of equal sides are equal or supplementary angles. In the former case, the two triangles are congruent. $$…
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Katayama-hiko Shrine (1873), Osafune-cho, Setouchi City, Okayama Prefecture (01)
Katayama-hiko Shrine is located $850$ meters southeast from Osafune Station on the JR Ako Line. Problem As shown in the figure, two mutually circumscribed circles of diameter $d$ are inscribed in a fan of radius $R$. $c$ is the common tangent of these two circles, and is also the chord of the fan that inscribes…
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The Encyclopedia of Geometry (0034)
Problem Two triangles which have two equal sides and equal median lines are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $AB=DE$, $AC=DF$ and $AP=DQ$. If we take points $A’$ and…
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The Encyclopedia of Geometry (0033)
Problem Two triangles with equal bases, vertex angles, and heights are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△ABC$ and $△A’B’C’$, suppose that $$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 $A$,…