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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|$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0057)
Problem Create an isosceles triangle $ALM$ that overlaps two sides $AB$ and $AC$ of a triangle $ABC$, extend $LM$ and $BC$, and let $N$ be their intersection. Then the straight line $LN$ intersects $AB$ at an angle equal to half the sum of its lower angles, and intersects the extension of the base of the…
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Katayama-hiko Shrine (1873), Osafune-cho, Setouchi City, Okayama Prefecture (05)
Problem As shown in the figure, there are four circles inscribed in a right triangle, and there are large, medium, and small red circles between them. If the diameter of the large circle is $4$ inches and the diameter of the middle circle is $2$ inches, find the diameter of the small circle. $$ $$…
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The Encyclopedia of Geometry (0056)
Problem In $△ABC$, if the bisector of $∠A$ is $AN$ and the perpendicular line from $A$ to $BC$ is $AH$, then $$∠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,$$…