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The Encyclopedia of Geometry (0173)
Problem In a triangle $ABC$, let $AC>AB$. Let the perpendicular line from $B$ to $AC$ be $BH$. Let the perpendicular lines from a point $P$ on $BC$ to $AB$ and $AC$ be $PE$ and $PD$, respectively. Then $$PD+PE>BH.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0172)
Problem There are two half lines $OX$ and $OY$ starting at $O$. Let a point $P$ be within $∠XOY$ and the feet of perpendicular lines drawn from it to $OX$ and $OY$ be $Q$ and $S$, respectively. Then, if the difference between $PS$ and $PQ$ is a constant $m$, then the point $P$ is always…
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The Encyclopedia of Geometry (0171)
Problem From a point $P$ in the given angle $∠XAY$, drop perpendicular lines $PQ$ and $PR$ to $AX$ and $AY$. If $m$ is a positive constant, then the point $P$ is on a fixed line segment such that $PQ+PR=m$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0170)
Problem If $D$ and $E$ are the points that trisect the side $BC$ of triangle $ABC$, then $$AB+AC>AD+AE.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If the midpoint of $BC$ is $M$, and $AM$ is extended to the…
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The Encyclopedia of Geometry (0169)
Problem From the midpoints $P$ and $Q$ of sides $AB$ and $AC$ of triangle $ABC$, draw perpendicular lines $PD$ and $QE$ to the outside of the triangle such that $$PD=\frac{1}{2} AB \qquad and \qquad QE=\frac{1}{2} AC.$$ Then, $DM$ and $EM$, which connect $D$ and $E$ to the midpoint $M$ of side $BC$, are equal and…
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The Encyclopedia of Geometry (0168)
Problem (1) Let $M$ be the midpoint of the line segment $AB$. Connect $M$ to a point $P$ outside this line. If $MP<MA$, which is $∠APB$ an acute or obtuse angle? Furthermore, what if $MP>MA$? (2) Prove that the midpoint of the hypotenuse of a right…
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The Encyclopedia of Geometry (0167)
Problem Let $D$ and $E$ be points on sides $BC$ and $CA$ respectively of a triangle $ABC$, such that $$BD=\frac{1}{2} DC \qquad and \qquad CE=EA.$$ Then $AD$ bisects $BE$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $F$…
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The Encyclopedia of Geometry (0165)
Problem Take a point $P$ in an equilateral triangle $ABC$. Let $D, \ E$ and $F$ be the feet of perpendicular lines drawn from $P$ to sides $BC, \ AB$ and $CA$, respectively. When $P$ is on the line segment joining the midpoints of $AB$ and $AC$, prove that $PD=PE+PF$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$…
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The Encyclopedia of Geometry (0164)
Problem The sum of the lengths of the perpendiculars $PE, \ PF$ and $PG$ from any point $P$ in an equilateral triangle $ABC$ to sides $BC, \ CA$ and $AB$ is constant. Moreover, what if $P$ is outside the equilateral triangle $ABC$? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0163)
Problem Draw equilateral triangles $ABP$ and $CDR$ with opposite sides $AB$ and $CD$ on the outside of the quadrilateral $ABCD$. When drawing an equilateral triangle with side $BC$ as the base inside the quadrilateral, $$PQ=AC \qquad and \qquad QR=BD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…