Japan “Sangaku” Research Institute

Problem Let $E$ be a point on diagonal $BD$ of square $ABCD$ such that $BE=BC$. From any point $P$ on segment $CE$, drop perpendiculars to $BD$ and $BC$, and let $F$ and $G$ be the respective feet. Then $$PF+PG=\frac{1}{2} BD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Let rectangle $PQRS$ be inscribed in square $ABCD$. Each side of the rectangle is parallel to a diagonal of the square. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution   Let $O$ be the intersection point of the…

Problem Draw squares $ABEF$ and $ACGH$ externally on triangle $ABC$. On the same side of the triangle, construct an isosceles right triangle $BCP$ with $BC$ as its hypotenuse. Then points $E, \ P$ and $G$ are collinear, and $$EP=PG.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

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? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…