Japan “Sangaku” Research Institute

Problem In square $ABCD$, let $A$ be connected to a point $E$ on side $BC$. Let $F$ denote the intersection of the bisector of $∠EAD$ with side $CD$. Then $$DF=AE-EB.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let…

Problem If points $E$ and $F$ are chosen on opposite sides $AB$ and $CD$ of square $ABCD$, respectively, and a line perpendicular to $EF$ intersects $AD$ and $BC$ (or their extensions) at points $H$ and $G$, then $$EF = HG.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If perpendicular lines $BE$ and $DF$ are drawn from $B$ and $D$ to any line $XY$ that passes through the vertex $C$ of square $ABCD$, then $$DF+BE=EF \qquad or \qquad |DF-BE|=EF.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Five people, $A, \ B, \ C,  D$ and $E$, decided to invest a total of $1,560 \ yen$. $B$ invested $30 \ yen$ less than $A$, $C$ invested $50 \ yen$ less than $B$, $D$ invested $80 \ yen$ less than $C$, and $E$ invested $110 \ yen$ less than $D$. Find the…

Problem If the feet of the perpendiculars drawn from the opposite vertices $A$ and $C$ of the square $ABCD$ to any line passing through the other vertex ($B$ or $D$) are $A’$ and $C’$, respectively, then $$AA’=BC’.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If we take an arbitrary $G$ on the side $DC$ of a square $ABCD$ and draw the square $GCEF$ outside the $ABCD$ with one side being $CG$, then $$DE⊥BG \qquad and \qquad DE=BG.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If we take the point $E$ on the diagonal $BD$ of a square $ABCD$ so that $AB=BE$, and the perpendicular line passing through that point intersects $CD$ at the point $F$, then $$CF=ED.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…