Japan “Sangaku” Research Institute

The Encyclopedia of Geometry (0028)

Problem Two triangles in which a pair of corresponding sides and the included angle are equal are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Now, if we put $A$ on…

February 2, 2024
The Encyclopedia of Geometry (0027)

Problem When two line segments $AB$ and $CD$ are parallel to each other in the same direction, if we take a point $P$ that is not between $AB$ and $CD$, then $$|∠ABP-∠CDP|=∠BPD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

February 1, 2024
The Encyclopedia of Geometry (0026)

Problem Suppose that two line segments $AB$ and $CD$ are parallel to each other in the same direction. Then, if the point $P$ is between $AB$ and $CD$, $$∠ABP+∠CDP=∠BPD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

January 30, 2024
Koh-jingu Shrine (1880), Higashiaso, Soja City, Okayama Prefecture (03)

Problem If a person’s moon shadow is reflected in the water, its length is $10 ft$ from the feet, and the height from the water’s surface to the eyes is $6 ft$, How many degrees does the moon hang over? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$…

January 29, 2024
The Encyclopedia of Geometry (0025)

Problem If the two sides $AB$ and $AC$ of $∠BAC$ are, respectively, parallel to the two sides $DE$ and $DF$ of $∠EDF$, then the bisector $AM$ of the former angle and the bisector $DN$ of the latter are parallel or perpendicular to each other. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$…

January 28, 2024
The Encyclopedia of Geometry (0024)

Problem When a straight line $EF$ cuts across parallel lines $AB$ and $CD$, the bisectors of the alternate angles and the corresponding angles are parallel. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…

January 26, 2024
The Encyclopedia of Geometry (0023)

Problem For $∠ABC$ and $∠DEF$, if the sides $AB$ and $DE$ are parallel in the same direction, and $BC$ and $EF$ are parallel in opposite direction, then $$∠ABC+∠DEF=2∠R.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

January 25, 2024
Koh-jingu Shrine (1880), Higashiaso, Soja City, Okayama Prefecture (02)

Problem If the distance of $1°$ at the equator is $28.03 \ ri$, how many degrees north latitude is the place where the distance of $1°$ is $27 \ ri$ ? ($1 \ ri$ is approximately $3.927 \ km$.) $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$…

January 22, 2024
The Encyclopedia of Geometry (0022)

Problem When the two sides $AB$ and $BC$ of $∠ABC$ are, respectively, parallel to the two sides $DE$ and $EF$ of $∠DEF$ in the same direction or in the opposite direction, then $$∠ABC=∠DEF.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

January 21, 2024
The Encyclopedia of Geometry (0021)

Problem When there are two straight lines that intersect on the same plane, two straight lines that are parallel to each of them will always intersect. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

January 20, 2024