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The Encyclopedia of Geometry (0143)
Problem Let the vertex angle $A$ of an isosceles triangle $ABC$ be $120°$. Let $D$ be the foot of the perpendicular line drawn from the vertex angle $A$ to $BC$, and take any point $P$ on $AD$ and connect it to $B$ and $C$. The following inequality holds: $$AP+BP+CP>AB+AC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…
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The Encyclopedia of Geometry (0142)
Problem If the bisectors of $∠B$ and $∠C$ of a triangle $ABC$ are equal, then the triangle is isosceles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If we assume that $∠B>∠C$, then from the problem $0080$,$$BD<CE.$$Similarly, if we…
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The Encyclopedia of Geometry (0141)
Problem The straight line connecting the intersection of perpendicular lines drawn from both ends of the base of an isosceles triangle to the vertex bisects the apex angle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If the intersection…
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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…