Japan “Sangaku” Research Institute

Asuhayama Shrine (1875b), Ibara-cho, Ibara City, Okayama Prefecture (01)

Asuhayama Shrine is located 750 $m$ north of Ibara Station on the Ibara Line of the Ibara Railway. Problem As shown in the figure, the side $BC$ is one side of a regular pentagon, which has a diagonal $DF$. When we know the lengths of sides $AB$ and $AC$ and the angle $∠A$, find…

November 14, 2024
The Encyclopedia of Geometry (0160)

Problem Divide the line segment $AB$ at $C$ and construct equilateral triangles $ACP$ and $BCQ$ on $AC$ and $BC$. Then, the sum of the heights of the two triangles is constant. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…

November 11, 2024
The Encyclopedia of Geometry (0159)

Problem $O$ is a point in an equilateral triangle $ABC$. If $∠BAO>∠CAO$, then $$∠BCO>∠CBO.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution $△ABO$ and $△ACO$ share the side $AO$,$$AB=AC \qquad and \qquad ∠BAO>∠CAO,$$$$∴ \ BO>CO,$$$$∴ \ ∠BCO>∠CBO.$$ $…

November 9, 2024
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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

November 5, 2024
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…

November 3, 2024
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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$…

October 31, 2024
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 $CN$…

October 27, 2024
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…

October 24, 2024
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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

October 21, 2024
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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

October 19, 2024