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Eiryu-ji Temple (1864), Shiokawa Town, Suzaka City, Nagano Prefecture (01)
Eiryu-ji Temple is said to be located in Shiokawa-machi, Suzaka City, Nagano Prefecture, but its exact location is unknown. Problem As shown in the figure, consider a right‑angled triangle $ABC$ with $BC=a$ (constant) and $AC=x$ (variable). Draw an arc of radius $x$ centered at vertex $A$. Let $y$ be the side length of the square…
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Formula 002
[Formula] Let $a$ denote the length of side $AB$ in the square shown above. Then $$AC=\sqrt{2} a.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ [Proof] $$AC=\sqrt{AB^2+BC^2}=\sqrt{a^2+a^2}=\sqrt{2a^2}=\sqrt{2}a.$$ Alternatively, using the ratio of the side and…
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The Encyclopedia of Geometry (0287)
Problem The perpendicular from the center $O$ of the circle to the chord $AB$ bisects both the major and minor arcs determined by the chord. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $H$ and $K$ be the…
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The Encyclopedia of Geometry (0286)
Problem A straight line drawn from the center of a circle to the midpoint of a chord is perpendicular to that chord. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $O$ be the center of the circle,…
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Formula 001
[Formula] Let $a$ be the length of side $AB$ of the equilateral triangle $ABC$ shown above, and let $O$ be the centroid of the triangle. Then $$AH=\frac{\sqrt{3}}{2}a,$$ $$CO=\frac{1}{\sqrt{3}}a,$$ $$HO=\frac{1}{2\sqrt{3}}a.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0285)
Problem Let $C_1, \ C_2$, and $C_3$ be three circles with radii $r_1, \ r_2$, and $r_3$, respectively. Denote the distances between the centers of $C_2$ and $C_3$, $C_3$ and $C_1$, and $C_1$ and $C_2$ by $d_{23}, \ d_{31}$, and $d_{12}$, respectively. Assume that the following inequalities hold: $$d_{12}+r_2<r_1,$$ $$d_{31}+r_3<r_1,$$ $$r_2+r_3<d_{23}.$$ What is the relative…
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The Encyclopedia of Geometry (0284)
Problem Given two circles with radii $r$ and $r′ \ (r>r′)$ and a distance $d$ between their centers, describe the relationships among $r, \ r′$, and $d$ for the various possible relative positions of the circles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0283)
Problem Consider an annulus whose inner radius is $r$, where $r$ is half the radius of the outer circle. When the annulus rolls one full revolution without slipping along a straight line, how many times does the inner circle rotate, and how far does it slip relative to the ground? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$…
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Suwa Taisha Shimosha Akimiya (1868), Shimosuwa Town, Suwa District, Nagano Prefecture (01)
Suwa Taisha Shimosha Akimiya is located about $800 \ meters$ northeast of JR Shimosuwa Station. Problem As shown in the figure, a square is inscribed in a rhombus. When the side length of the square is $\frac{325}{38} \ inches$, find the lengths of the rhombus’s vertical and horizontal diagonals. $$ $$ $$ $$ $\downarrow$ $\downarrow$…