Japan “Sangaku” Research Institute

Problem Take a point $P$ in an equilateral triangle $ABC$. Let $D, \ E$ and $F$ be the feet of perpendicular lines drawn from $P$ to sides $BC, \ AB$ and $CA$, respectively. When $P$ is on the line segment joining the midpoints of $AB$ and $AC$, prove that $PD=PE+PF$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$…

Problem The sum of the lengths of the perpendiculars $PE, \ PF$ and $PG$ from any point $P$ in an equilateral triangle $ABC$ to sides $BC, \ CA$ and $AB$ is constant. Moreover, what if $P$ is outside the equilateral triangle $ABC$? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Draw equilateral triangles $ABP$ and $CDR$ with opposite sides $AB$ and $CD$ on the outside of the quadrilateral $ABCD$. When drawing an equilateral triangle with side $BC$ as the base inside the quadrilateral, $$PQ=AC \qquad and \qquad QR=BD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If each side of a triangle $ABC$ is used as base and equilateral triangles $BCD, \ CAE$ and $ABF$ are constructed outside the triangle, then the lengths of line segments $AD, \ BE$ and $CF$ are equal. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If equilateral triangles $BCX$ and $CDY$, with the bases $BC$ and $CD$ of a parallelogram $ABCD$, are drawn outside the quadrilateral, then $△AXY$ becomes an equilateral triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△ABX$ and…

Problem If we take a point $C$ on the extension of a line segment $AB$ (or $BA$) and construct equilateral triangles $ACP$ and $BCQ$ on $AC$ and $BC$, the absolute value of the difference of their heights is constant. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

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…

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 $AO$,$$AB=AC \qquad and \qquad ∠BAO>∠CAO,$$$$∴ \ BO>CO,$$$$∴ \ ∠BCO>∠CBO.$$ $…