Japan “Sangaku” Research Institute

Hioki Shrine (1911), Shinshu-shinmachi, Nagano City, Nagano Prefecture (05)

Problem As shown in the figure, the square $BDEF$ is inscribed in the right triangle $ABC$, and the sum of the areas of $⊿ADE$ and $⊿EFC$ is $150 \ in^2$, and the length of the side $AB$ is $21 \ in$. Find the length of one side of the square $BDEF$. $$ $$ $$ $$…

August 26, 2025
The Encyclopedia of Geometry (0214)

Problem The two diagonals of a rhombus are perpendicular to each other and bisect each vertex angle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution $△ABD$ and $△CDB$ share $BD \ (=DB)$, and $$AB=CD \qquad and \qquad AD=CB,$$ $$∴ \quad…

August 19, 2025
The Encyclopedia of Geometry (0213)

Problem The two diagonals of a rectangle are equal. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△ABC$ and $△DCB$,$$AB=DC, \qquad BC=CB \qquad and \qquad ∠B=∠C \ (=∠R),$$$$∴ \quad △ABC≡△DCB,$$$$∴ \quad AC=DB.$$ $ $ $ $ $…

August 12, 2025
Hioki Shrine (1911), Shinshu-shinmachi, Nagano City, Nagano Prefecture (04)

Problem You purchased $400$ pieces of lumber for $40$ dollars. The pieces were high-quality at $0.16$ dollars each and low-quality at $0.08$ dollars each. How many pieces of high-quality lumber did you buy at this time? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

August 10, 2025
The Encyclopedia of Geometry (0212)

Problem Within a parallelogram $ABCD$, draw $OA, \ OB, \ OC$, and $OD$ by connecting any point $O$ to each of the vertices, and let the midpoints of each be $E, \ F, \ G$, and $H$. Then, if the intersections of $DE$ and $CF$, $AF$ and $DG$, $AH$ and $BG$, and $BE$ and $CH$…

August 9, 2025
Hioki Shrine (1911), Shinshu-shinmachi, Nagano City, Nagano Prefecture (03)

Problem There are three squares: large, medium, and small. The sum of the areas of the three squares is $133 \ in^2$. The medium square is $20 \ in^2$ larger than the small square, and the large square is $45 \ in^2$ larger than the medium one.Find the length of one side of the small…

August 2, 2025
Aoki Hachiman Shrine (1854), Nishinoura, Tsurajima-cho, Kurashiki City, Okayama Prefecture (01)

According to Yamakawa, the location of Aoki Hachiman Shrine is Aoki, Nishinoura, Tsurajima-cho, Kurashiki City, Okayama Prefecture, but this cannot be confirmed. Problem As shown in the figure, there are two large circles and one small circle with their centers on a line, and the small circle is between the square and the large circles.If…

July 28, 2025
The Encyclopedia of Geometry (0211)

Problem If equilateral triangles $ABD$ and $ACE$ are drawn outside a triangle $ABC$, and an equilateral triangle $BCF$ with the side $BC$ is drawn on the same side as a vertex $A$, then the quadrilateral $AEFD$ is a parallelogram. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

July 25, 2025
The Encyclopedia of Geometry (0209)

Problem When you draw four squares with each side as one side on the outside of the parallelogram $ABCD$ and connect the centers of these squares to form a quadrilateral, it will be a square. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

July 14, 2025
The Encyclopedia of Geometry (0210)

Problem If you draw the equilateral triangles $ABE$ and $CDF$ on the outside of a parallelogram $ABCD$ and the equilateral triangle $BCG$ on the same side as the parallelogram, then $$EG=AC \qquad and \qquad FG=DB.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

July 5, 2025