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The Encyclopedia of Geometry (0158)
Problem Take three points $D, \ E$ and $F$ on each side of a triangle $ABC$ such that $AD=BE=CF$. Then, if the triangle $DEF$ is an equilateral triangle, $△ABC$ is also an equilateral triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0157)
Problem If there are three points $D, \ E$ and $F$ on each side of an equilateral triangle $ABC$ such that $AD=BE=CF$, then the triangle $DEF$ is also an equilateral triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…
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The Encyclopedia of Geometry (0156)
Problem Let $DE$ and $DF$ be the perpendicular lines drawn from any point $D$ on the hypotenuse $BC$ of a right isosceles triangle $ABC$ to the sides $AB$ and $AC$, respectively, and let $M$ be the midpoint of BC. Then $△EMF$ is also an isosceles right triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$…
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The Encyclopedia of Geometry (0155)
Problem Of the two points of intersection between the two lines that trisect $∠B$ of an isosceles triangle $ABC$ and the perpendicular line $AD$ drawn from the vertex $A$ to the base $BC$, the one closer to $A$ is called $M$ and the other is called $N$. If the intersection point of the line $CN$…
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The Encyclopedia of Geometry (0154)
Problem Let $E$ and $F$ be the points where the line that passes through a point $D$ on the base $BC$ of an isosceles triangle $ABC$ and is perpendicular to $BC$ intersects with $AB$ and $AC$ (or their extensions). Construct rectangles $EDBG$ and $FDCH$, and let $S$ and $T$ be the midpoints of $DG$ and…
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The Encyclopedia of Geometry (0153)
Problem If perpendicular lines $PD$ and $PE$ are dropped from any point $P$ on the base $BC$ of an isosceles triangle $ABC$ with $A$ as its vertex to the sides $AB$ and $AC$, then $AD+AE$ is constant. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0152)
Problem The difference between the perpendicular lines $PQ$ and $PR$ drawn from any point $P$ on the extension of the base $BC$ of an isosceles triangle $ABC$ to the sides $AB$ and $AC$ (or their extensions) is constant. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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Katayama-hiko Shrine (1873), Osafune-cho, Setouchi City, Okayama Prefecture (16)
Problem As shown in the figure, there is an equilateral triangle, one large circle, two medium circles, and one small circle within a square. If the diameter of the large circle is $9.5$ inches, find the diameter of the medium circle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$…
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Katayama-hiko Shrine (1873), Osafune-cho, Setouchi City, Okayama Prefecture (15)
Problem A circle is inscribed in an isosceles triangle with two equal sides of $10 \ inches$ and a base of $12 \ inches$.What is the diameter of the inscribed circle? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0151)
Problem The sum of perpendicular lines $PQ$ and $PR$ drawn from any point $P$ on the base $BC$ of an isosceles triangle $ABC$ with $A$ as its vertex to $AB$ and $AC$ is constant. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…