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The Encyclopedia of Geometry (0052)
Problem Create two triangles $△ABC$ and $△DBC$ on the same side of the same base $BC$. Suppose that $$AB+AC=DB+DC, \quad AB=AC \quad and \quad DB>DC.$$ If the intersection of $DB$ and $AC$ is $E$, then $$AE>DE.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0051)
Problem In $△ABC$, when $∠B=2∠C$, $$AC<2AB.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If we take $E$ on $BC$ so that $∠CAE=∠C$, $$EA=CE.$$ Since $∠AEB$ is the exterior angle of $△EAC$, $$∠AEB=∠CAE+∠C,$$…
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The Encyclopedia of Geometry (0050)
Problem If we take any point $F$ inside $△ABC$ and any two points $H$ and $I$ on $BC$, will the proposition $AB+AC>FH+FI$ always hold? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If…
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The Encyclopedia of Geometry (0049)
Problem In the obtuse triangle $ABC$, let $∠C$ be the obtuse angle.Suppose that$$BD=\frac{1}{2} (AB+AC), \qquad and \qquad DE∥BC.$$Take any point $F$ inside $△ADE$ or on $DE$, draw a straight line through $F$ and parallel to $AB$, and let $G$ be the point of intersection with $BC$. If we take $H$ and $I$ on the side…
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The Encyclopedia of Geometry (0048)
Problem In $△ABC$, when $BC<AC<AB$, take any point $P$ in the triangle, create line segments $AP$, $BP$, and $CP$, and let $M$ be the intersection of $BC$ and the extension of $AP$, then, $$(1) \ AM+BC<AB+AC.$$ $$(2) \ AP+BP+CP<AB+AC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0047)
Problem If we take an arbitrary point $O$ in $△ABC$ and represent $BC, \ CA$, and $AB$ by $a, \ b$, and $c$, respectively, we get $$a+b+c>AO+BO+CO>\frac{1}{2} (a+b+c).$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0046)
Problem If we take any point $O$ within $△ABC$ and draw line segments $OB$ and $OC$, we get $$AB+AC>OB+OC, \qquad and \qquad ∠BOC>∠A.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If $BO$…
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The Encyclopedia of Geometry (0045)
Problem If we take any points $D, \ E$, and $F$ on sides $BC, \ CA$, and $AB$ of $△ABC$, we get $$3(AD+BE+CF)<5(AB+BC+CA).$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If $AB$…
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The Encyclopedia of Geometry (0044)
Problem When we take any two points $P$ and $Q$ inside or on the circumference of $△ABC$, the length of line segment $PQ$ does not exceed the length of the maximum side of this triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0043)
Problem In $△ABC$, when $AC$ is larger than $AB$, $AD$ connecting any $D$ on $BC$ and $A$ is smaller than $AC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If $AC>AB$, then from…