Japan “Sangaku” Research Institute

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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem When the tangent angles $∠AOC$ and $∠BOC$ are complementary angles to each other, let the bisectors of $∠AOC$ and $∠BOC$ be $OD$ and $OE$ respectively. Then,  $$OD⊥OE.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem “If there are straight lines $A’O’$ and $B’O’$ that both pass through point $O’$, and $A’O’$ and $B’O’$ are respectively perpendicular to straight lines $AO$ and $BO$ that both pass through point $O$, then $∠AOB$ and $∠A’O’B’$ are equal.” Is this proposition correct? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$…

Problem If there are two angles $∠AOB$ and $∠COD$ with the same vertex, and $AO⊥CO$ and $BO⊥DO$, then $∠AOB=∠COD \quad$ or $\quad ∠AOB+∠COD=2∠R$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution In figure…

Problem The opposite angles are equal. That is, when two straight lines $AB$ and $CD$ intersect at point $O$, $∠AOC = ∠BOD$     and     $∠AOD = ∠BOC$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…