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The Encyclopedia of Geometry (0066)
Problem In a triangle $ABC$, if $AB>AC$, and we take any point $P$ on the perpendicular $AD$ drawn from the vertex $A$ to the opposite side $BC$, then we have $$PB-PC>AB-AC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0065)
Problem Suppose that any straight line passes through the vertex $A$ of a triangle $ABC$. The feet $D$ and $E$ of the perpendicular lines drawn from $B$ and $C$ to the above line are equidistant from the midpoint $M$ of the side $BC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$…
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The Encyclopedia of Geometry (0064)
Problem If the feet of the perpendicular lines drawn from the two vertices $B$ and $C$ of a triangle $ABC$ to the opposite sides $AC$ and $AB$ are $E$ and $F$, respectively, then the straight line connecting the midpoint of the line segment $EF$ and the midpoint of the side $BC$ is perpendicular to $EF$.…
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The Encyclopedia of Geometry (0063)
Problem The length of the bisector of one angle of a triangle is less than the average length of the two sides forming that angle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…
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The Encyclopedia of Geometry (0062)
Problem The bisector $AE$ of the apex angle $∠A$ of a triangle $ABC$ lies between the median line $AM$ and the perpendicular $AD$ drawn from this apex angle to the opposite side. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0061)
Problem In $△ABC$, suppose $AC>AB$, and if we draw the perpendicular $AD$ from $A$ to $BC$, we have $$∠DAC>∠DAB \qquad and \qquad DC>DB$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Since $AD⊥BC$,…
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The Encyclopedia of Geometry (0060)
Problem The sum of the lengths of perpendiculars drawn from the three vertices of a triangle to their opposite sides is less than the sum of the lengths of the three sides of the triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0059)
Problem The angle made by perpendicular lines drawn from two vertices of an acute triangle to their opposite sides is equal to the supplementary angle of the remaining vertex. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0058)
Problem Let $D$ be the foot of the perpendicular drawn from the vertex $A$ to the opposite side $BC$ of $△ABC$, and let $E$ and $F$ be the midpoints of the sides $BC$ and $AB$, respectively. Then $$∠DFE=|∠B-∠C|$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0057)
Problem Create an isosceles triangle $ALM$ that overlaps two sides $AB$ and $AC$ of a triangle $ABC$, extend $LM$ and $BC$, and let $N$ be their intersection. Then the straight line $LN$ intersects $AB$ at an angle equal to half the sum of its lower angles, and intersects the extension of the base of the…