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The Encyclopedia of Geometry (0098)
Problem Let $AD, \ BE$ and $CF$ be the three median lines of $△ABC$, respectively. Then, $$\frac{1}{2} (AB+BC+CA)<AD+BE+CF<AB+BC+CA.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution From the problem $0097 \ (2)$, $$\frac{1}{2} (AB+CA-BC)<AD, \qquad \frac{1}{2} (AB+BC-CA)<BE,$$ $$and \qquad…
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The Encyclopedia of Geometry (0097)
Problem If the midpoint of the side $BC$ of $△ABC$ is $D$, then $$(1) \ AD<\frac{1}{2} (AB+AC).$$ $$(2) \ AD>\frac{1}{2} (AB+AC-BC).$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution $(1)$ If we extend $AD$ and take a point $E$…
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The Encyclopedia of Geometry (0096)
Problem In $△ABC$, if $∠B=60°$ and the points where the bisectors of $∠A$ and $∠C$ intersect with $BC$ and $AB$ are $D$ and $E$, respectively, $$CD+AE=AC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $F$ be the intersection…
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The Encyclopedia of Geometry (0095)
Problem The feet $P, \ Q, \ R$, and $S$ of the perpendiculars drawn from the vertex $A$ of $△ABC$ to the bisectors of the interior and exterior angles of the vertices $B$ and $C$ are on the same line. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0094)
Problem Let $D$ and $E$ be the feet of the perpendiculars drawn from $C$ to the bisectors of $∠A$ of $△ABC$ and its exterior angle, respectively, and let $M$ be the midpoint of the side $BC$. Then, $M, \ D$ and $E$ are on the same straight line. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…
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The Encyclopedia of Geometry (0093)
Problem Let $P$ and $Q$ be the feet of the perpendiculars drawn from $A$ to the bisectors of $∠B$ and $∠C$ of $△ABC$, respectively. Then, $$PQ=\frac{1}{2} (AB+AC-BC).$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let the intersections of…
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The Encyclopedia of Geometry (0092)
Problem Let $O$ be the intersection of the bisectors of $∠B$ and $∠C$ of $△ABC$, and let $D$ and $E$ be the feet of the perpendicular lines drawn from $A$ to the straight lines $BO$ and $CO$. Then $$DE∥BC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0091)
Problem In $△ABC$, if we draw perpendiculars $BG$ and $CH$ from $B$ and $C$ to the bisector of the exterior angle of $∠A$, and let $AD$ be the bisector of $∠A$, then $CG, \ BH$ and $AD$ pass through the same point. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0090)
Problem Let $E$ and $F$ be the feet of the perpendiculars drawn from both ends $B$ and $C$ of the base to the bisector of the apex angle $∠A$ of $△ABC$, and let $G$ be the midpoint of $BC$. Then, $△GEF$ is an isosceles triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0089)
Problem Suppose that the side $AB$ of $△ABC$ is one third of the side $AC$. Drop the perpendicular line $CF$ from $C$ to the bisector $AD$ of $∠A$. Then, $$AD=DF.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let…