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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…
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The Encyclopedia of Geometry (0056)
Problem In $△ABC$, if the bisector of $∠A$ is $AN$ and the perpendicular line from $A$ to $BC$ is $AH$, then $$∠HAN=\frac{1}{2} |∠B-∠C|$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution When $AB>AC$,…
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The Encyclopedia of Geometry (0055)
Problem In $△ABC$ where $AB>AC$, if we take $AD$ equal to $AC$ on $AB$, we have $$∠ADC=\frac{1}{2} (∠B+∠C) \quad and\quad ∠BCD=\frac{1}{2} (∠C-∠B)$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If $∠ADC=∠ACD$ and…
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The Encyclopedia of Geometry (0054)
Problem When extending any side of a triangle, its exterior angle is greater than any of its inner opposite angles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution From the previous question $0053$,…
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The Encyclopedia of Geometry (0053)
Problem The sum of the interior angles of a triangle is equal to two right angles, and the exterior angle is equal to the sum of its inner opposite angles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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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…