Sangaku Quest

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     $∠AOD = ∠BOC$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem There is one and only one straight line that passes through a point $O$ on a straight line $AB$ and is perpendicular to $AB$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…

Problem Prove that all flat angles (two right angles) are equal to each other. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution   Let the two flat angles be $∠BAC$ and $∠EDF$. Since…