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The Encyclopedia of Geometry (0078)
P roblem Is the statement “The bisector of the apex angle of a triangle and the perpendicular bisector of the opposite side intersect at a point outside the triangle” correct? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution The…
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The Encyclopedia of Geometry (0077)
P roblem In a triangle $ABC$, the angle formed by the intersection $E$ of the bisectors of $∠B$ and the exterior angle of $∠C$ is equal to half of $∠A$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let…
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The Encyclopedia of Geometry (0076)
Problem In a triangle $ABC$, let $O$ be the intersection of the bisectors of $∠B$ and $∠C$, and let $M$ and $N$ be the intersections of the straight line passing through $O$ and parallel to $BC$, and $AB$ and $AC$, respectively. Then, $$MN=|MB-NC|.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$…
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The Encyclopedia of Geometry (0075)
Problem Let $O$ be the intersection of the bisectors of $∠B$ and $∠C$ of a triangle $ABC$, and let $M$ and $N$ be the intersections of $AB$ and $AC$ with a straight line drawn through $O$ parallel to $BC$, respectively. Then, $$MN=MB+NC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0072)
Problem In a triangle $ABC$, let $∠B<∠C$ and draw the perpendiculars $BD$ and $CE$ from $B$ and $C$ to their opposite sides, respectively. Then, we have $$CE>BD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0074)
Problem Let $O$ be the intersection of the bisectors of $∠B$ and $∠C$ of a triangle $ABC$, and let $D$ be the intersection of the extension of $AO$ and $BC$. If the perpendicular from $O$ to $BC$ is $OE$, then $$∠BOE=∠COD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0073)
Problem If $D$ is the point where the bisector of $∠A$ intersects the side $BC$ in the triangle $ABC$, then $$AB>BD \qquad and \qquad AC>CD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…
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The Encyclopedia of Geometry (0071)
Problem If the feet of the perpendicular lines drawn from $A$ and $B$ to the opposite sides of a triangle $ABC$ are $D$ and $E$, respectively, and the midpoint of $AB$ is $F$, then $$∠EDF=∠C.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0070)
Problem In an acute triangle $ABC$, take points $P$ and $Q$ on the perpendicular lines drawn from the vertices $B$ and $C$ to the opposite sides, or on their extensions, so that $BP=CA$ and $CQ=BA$, respectively. Also, if we take points $P’$ and $Q’$ on $BC$ so that $PP’⊥BC$ and $QQ’⊥BC$, then $$PP’+QQ’=BC$$ $$ $$…
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The Encyclopedia of Geometry (0069)
Problem In a triangle $ABC$, let $∠C=2∠B$. Then, if we draw a perpendicular line $AD$ from $A$ to $BC$, the difference between $DB$ and $DC$ is equal to $AC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…