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The Encyclopedia of Geometry (0140)
Problem The medians drawn from both ends of the base of an isosceles triangle to the opposite sides are equal. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution $△DBC$ and $△ECB$ share the side $BC \ (=CB)$, $$DB=EC \qquad…
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The Encyclopedia of Geometry (0139)
Problem If the bisector of the vertex angle $A$ of a triangle $ABC$ is perpendicular to the base $BC$, then the triangle is an isosceles triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If the intersection point of…
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The Encyclopedia of Geometry (0138)
Problem In a triangle $ABC$, if $AB=AC$, then $$∠B=∠C.$$ And vice versa. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If we turn $△ABC$ over and make it $△AC’ B’$, $$AB=AC’ \qquad and \qquad AC=AB’, \qquad (∵ \ AB=AC)$$…
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The Encyclopedia of Geometry (0137)
Problem If we draw squares $BCDE$ and $ACFG$ with the sides $BC$ and $AC$ of a right-angled triangle $ABC$ with $∠B=∠R$ outside the triangle, and the intersection point of the lines $DF$ and $BC$ is $H$, then$$AB=2CH.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0136)
Problem In a right isosceles triangle $ABC$ with $∠B=∠R$, let $D$ and $E$ be the feet of the perpendicular lines drawn from a point $P$ on hypotenuse $AC$ to $AB$ and $AC$, respectively, and $F$ be the foot of the perpendicular line drawn from $P$ to $DE$. Then, the line $PF$ always passes through a…
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The Encyclopedia of Geometry (0135)
Problem Construct squares $ABDE$ and $ACFG$ with the sides $AB$ and $AC$ of a right triangle $ABC$ with $∠B=∠R$, and let $H$ be the intersection point between the extension of $BA$ and $EG$. Then, $$BC=2AH.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0134)
Problem Take a point $D$ inside the isosceles right triangle $ABC$ with $∠C=∠R$, so that $AD=AC$ and $∠CAD=30°$. Then $$BD=CD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If the point symmetrical to $A$ with respect to $DC$ is…
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The Encyclopedia of Geometry (0133)
Problem In a rectangular triangle $ABC$ with $B$ as the right-angled vertex, if $D$ and $E$ are the feet of perpendicular lines drawn from $A$ to the bisectors of $∠C$ and its exterior angle, then $DE$ bisects $AB$ vertically. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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Katayama-hiko Shrine (1873), Osafune-cho, Setouchi City, Okayama Prefecture (14)
Problem A regular pentagon is inscribed in a circle with a diameter of $1 \ ft$.Find the length of one side of this regular pentagon. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…
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The Encyclopedia of Geometry (0132)
Problem In a rectangular triangle $ABC$, let $F$ be the intersection point of the perpendicular line $BE$ drawn from the right-angled vertex $B$ to $AC$ and the bisector $AD$ of $∠A$, and let $G$ be the intersection point of $BC$ with a line passing through $F$ and parallel to $AC$. Then,$$BF=BD=GC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$…