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The Encyclopedia of Geometry (0121)
Problem In a triangle $ABC$ with right angle $∠B$, when $∠C=60°$ and $∠A=30°$, the hypotenuse $AC$ is twice the length of the side $BC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Extend $CB$ and take point $D$ such…
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Katayama-hiko Shrine (1873), Osafune-cho, Setouchi City, Okayama Prefecture (09)
Problem There are two large circles and two small circles inside a rhombus as shown in the figure. If the longer diagonal of the rhombus is $85 \ inches$ and the shorter is $42 \ inches$, find the diameters of the large and small circles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$…
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The Encyclopedia of Geometry (0120)
Problem If the vertex of the right angle of a right triangle $ABC$ is $C$ and the midpoint of the hypotenuse $AB$ is $D$, then $$CD=\frac{1}{2} AB.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If we extend $CD$…
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The Encyclopedia of Geometry (0119)
Problem There are two fixed points $A$ and $B$, and a moving point $P$ outside the line $AB$. Let $Q$ be the midpoint of $AP$ and $R$ be the midpoint of $BQ$. Then, $PR$ always passes through a fixed point on $AB$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0118)
Problem If the midpoint of the side $BC$ of $△ABC$ is $D$ and $∠ABD+∠DAC=∠R$, what type of triangle is $△ABC$ ? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Draw a circle circumscribing $△ABC$, and let $E$ be the…
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The Encyclopedia of Geometry (0117)
Problem How many lines are equidistant from the three vertices of $△ABC$ ? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $P, \ Q$ and $R$ be the midpoints of $BC, \ CA$ and $AB$, respectively. Every line…
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The Encyclopedia of Geometry (0116)
Problem In $△ABC$, extend the median $AM$ so that $MD=BC$. If $∠AMC=60°$, then $$BD⊥BC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Taking the midpoint $E$ of $MD$, $$ME=MB \qquad and \qquad ∠BME=60°.$$ Thus, $△MBE$ is an equilateral triangle,…
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The Encyclopedia of Geometry (0115)
Problem In $△ABC$, let $AB>AC$, and take any point $P$ on the median line $AD$. Then, $$AB-AC>PB-PC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If we take a point $E$ on $AB$ so that $AC=AE$, $$AB-AC=AB-AE=EB.$$ Since $AB>AC$…
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The Encyclopedia of Geometry (0114)
Problem Two lines $x$ and $y$ intersect at point $O$. On each of these lines are equal-length segments $AB$ and $CD$, and the midpoints of $AC$ and $BD$ are $M$ and $N$, respectively. Then, the line $MN$ has a constant direction regardless of the positions of $AB$ and $CD$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…
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The Encyclopedia of Geometry (0113)
Problem Construct the median $AD$ of $△ABC$, and draw a line parallel to $AD$ from any point $P$ on the side $BC$. Let $Q$ and $R$ be the points where the line intersects with $AB$ and $AC$, or their extensions. Then, the length of $PQ+PR$ is constant regardless of the location of $P$. $$ $$ $$ $$ $\downarrow$…