Japan “Sangaku” Research Institute

Problem The two straight lines $EF$ and $GH$, which are perpendicular to the two parallel lines $AB$ and $CD$, respectively, are parallel to each other. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…

Problem When a straight line $XY$ is perpendicular to $CD$ of the parallel lines $AB$ and $CD$, it is also perpendicular to the other $AB$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…

Problem Two straight lines perpendicular to the same straight line are parallel to each other. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Assume that $AB$ and $CD$ are both orthogonal to $XY$…

Problem When three straight lines $a, b$ and $c$ are on the same plane, and $a∥b$ and $a∥c$, then $$b∥c.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Assuming $b∦c$, two straight lines…

Problem When two parallel lines intersect with a straight line, the corresponding angles or alternate angles are equal, and the interior angles (or the exterior angles) on the same side are complementary to each other. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Two straight lines are parallel when they intersect with a straight line and alternate angles they form are equal. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let the two straight lines…

Problem The point $P$ on the bisector of $∠BAC$ is equidistant from its two sides $AB$ and $AC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Drop a perpendicular line from the point…

Problem If the bisectors of tangent angles $∠AOB$ and $∠BOC$ are $OM$ and $ON$ respectively, then $$∠MON=\frac{1}{2}(∠AOB+∠BOC).$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Since $OM$ is the bisector of $∠AOB$, as…

Problem If the bisectors $OM$ and $ON$ of the tangent angles $∠AOB$ and $∠BOC$ are perpendicular to each other, $OA$ and $OC$ form a straight line. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…