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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, $$PQ=AC \qquad and \qquad QR=BD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0162)
Problem If each side of a triangle $ABC$ is used as base and equilateral triangles $BCD, \ CAE$ and $ABF$ are constructed outside the triangle, then the lengths of line segments $AD, \ BE$ and $CF$ are equal. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0161)
Problem If equilateral triangles $BCX$ and $CDY$, with the bases $BC$ and $CD$ of a parallelogram $ABCD$, are drawn outside the quadrilateral, then $△AXY$ becomes an equilateral triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△ABX$ and…
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The Encyclopedia of Geometry (0160_2)
Problem If we take a point $C$ on the extension of a line segment $AB$ (or $BA$) and construct equilateral triangles $ACP$ and $BCQ$ on $AC$ and $BC$, the absolute value of the difference of their heights is constant. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0160)
Problem Divide the line segment $AB$ at $C$ and construct equilateral triangles $ACP$ and $BCQ$ on $AC$ and $BC$. Then, the sum of the heights of the two triangles is constant. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…
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The Encyclopedia of Geometry (0159)
Problem $O$ is a point in an equilateral triangle $ABC$. If $∠BAO>∠CAO$, then $$∠BCO>∠CBO.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution $△ABO$ and $△ACO$ share the side $AO$,$$AB=AC \qquad and \qquad ∠BAO>∠CAO,$$$$∴ \ BO>CO,$$$$∴ \ ∠BCO>∠CBO.$$ $…