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The Encyclopedia of Geometry (0103)
Problem In $△ABC$, let $D$ be the midpoint of $AB$. Then, the length of the line segment connecting $D$ and the midpoint $E$ of $AC$ is $\frac{1}{2} BC$. State the converse of this theorem and determine the conditions for it to hold. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0102)
Problem Let $D$ be the midpoint of the side $AB$ of $△ABC$. If a circle with radius $\frac{1}{2} BC$ centered on $D$ is drawn, prove that it intersects with $AC$ at one or two points, and determine the conditions for there to be only one point of intersection. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…
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The Encyclopedia of Geometry (0101)
Problem Is the following statement correct ? Let the line passing through the midpoint $M$ of the side $AB$ of $△ABC$ intersect with the point $N$ on the side $AC$. In this case, if $MN=\frac{1}{2} BC$, then $$MN∥BC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0100)
Problem In $△ABC$, suppose $AB>AC$. Then, if we draw the median line $AD$, $$∠BAD<∠CAD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $E$ be the intersection point of the line parallel to $AC$ that passes through $B$ and…
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The Encyclopedia of Geometry (0099)
Problem Let $AD, \ BE$, and $CF$ be the three medians of $△ABC$, and $G$ be their intersection point. Then, $$AD+BE+CF<AB+BC+CA<2 \ (AG+BG+CG).$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution First, from the problem $0098$, $$AD+BE+CF<AB+BC+CA. \qquad [1]$$…
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The Encyclopedia of Geometry (0098)
Problem Let $AD, \ BE$ and $CF$ be the three median lines of $△ABC$, respectively. Then, $$\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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…