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The Encyclopedia of Geometry (0124)
Problem If $D$ is the foot of the perpendicular line drawn from the right-angled vertex $A$ of a rectangular triangle $ABC$ to the hypotenuse $BC$, then$$AD+BC>AB+AC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Take a point $E$ on…
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The Encyclopedia of Geometry (0123)
Problem If $D$ is the intersection point of the bisector of the right-angled vertex $A$ of a rectangular triangle $ABC$ and the line that passes through the midpoint $M$ of the hypotenuse $BC$ and is perpendicular to $BC$, then $$MA=MD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0122)
Problem If you drop a perpendicular line $AD$ from the right-angled vertex $A$ of a rectangular triangle $ABC$ to the side $BC$, $$∠C=∠BAD \qquad and \qquad ∠B=∠CAD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution $△ABC$ and $△DBA$ share…
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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,…