Sangaku Quest

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

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…