Japan “Sangaku” Research Institute

Problem Draw squares $ABDM$ and $ACEN$ externally on sides $AB$ and $AC$ of triangle $ABC$, respectively. If perpendiculars $DF$ and $EG$ are dropped from vertices $D$ and $E$ to line $BC$, then the length of $BC$ equals the sum of $DF$ and $EG$, and the area of triangle $ABC$ equals the sum of the areas…

Problem When a circle passing through the center $O$ and vertex $A$ of square $ABCD$ intersects sides $AB$ and $AD$ at points $P$ and $Q$, respectively, prove that $AP+AQ$ is equal to the side length of the square. Assume that $P$ and $Q$ lie on the sides (not on their extensions). $$ $$ $$ $$ $\downarrow$ $\downarrow$…

Problem There are two squares of the same size. If one square is placed with its center on a vertex of the other and then rotated, how does the area of the overlapping region change? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

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 $$EF = HG.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…