Japan SANGAKU Research Institute

Problem Assume that four moving points $P, \ Q, \ R$, and $S$ are initially located at the vertices $A, \ B, \ C$, and $D$ of square $ABCD$, respectively. Starting simultaneously, the points move along the sides $AB, \ BC, \ CD$, and $DA$ toward $B, \ C, \ D$, and $A$, all at…

Problem From vertex $A$ of square $ABCD$, draw a line parallel to the diagonal $DB$ and oriented in the same direction. Choose a point $E$ on this line such that $BE=BD$, and let $F$ be the point where line $EB$ meets the extension of $AD$. Then $$DF=DE.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$…

Problem Draw a line through vertex $B$ parallel to the diagonal $AC$ of square $ABCD$, and choose a point $F$ on this line such that $AF=AC$. Next, choose a point $E$ on the same line so that $CAFE$ forms a rhombus. Then the lines $AE$ and $AF$ trisect $∠BAC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…

Problem Let $E$ and $F$ be the points where a line parallel to the diagonal $AC$ of square $ABCD$ meets sides $AB$ and $BC$, respectively. Choose a point $G$ on the extension of side $DA$ beyond $A$ such that $AG=AD$, and let $H$ be the intersection of lines $GE$ and $DF$. Then $AH$ is equal…

Problem Take a point $E$ on side $AD$ of square $ABCD$ such that $BE=DE+DC$, and let $M$ be the midpoint of $AD$. Then $$∠EBC=2∠ABM.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Take a point $K$ on the extension…

Problem If we construct a square $ABDE$ on side $AB$ of triangle $ABC$, either externally or internally on the same side, and extend the perpendicular $AF$ from $A$ to $BC$ to a point $G$ on the side of $A$ (or $F$) such that $AG=BC$, then $$BG=CD.$$   $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$…