Japan “Sangaku” Research Institute

Problem If $F$ is the point at which the perpendicular line $CE$ drawn from the vertex $C$ of a rectangle $ABCD$ to the diagonal $BD$ intersects with the bisector of the angle $∠A$, then $$AC=CF.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If the quadrilateral formed by the intersection of the bisectors of each vertex angle of a quadrilateral is a square, what should the original quadrilateral be like? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $PQRS$ be…

Problem The diagonal of a rectangle is longer than any line segment drawn between its opposite sides. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $EF$ be any line segment drawn between $AB$ and $CD$. If we draw…

Problem If a parallelogram whose sides are parallel to the diagonals of a rectangle $ABCD$ is inscribed in the rectangle, the perimeter of the parallelogram will always be constant. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $K$…

Problem A quadrilateral formed by connecting the points where two straight lines that pass through the intersection of the diagonals of a parallelogram and intersect with each other at right angles intersect with each side is a rhombus. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If the intersection of the bisectors of the two vertex angles $B$ and $C$ of the rectangle $ABCD$ is $E$, and similarly, the intersections of the bisectors of $A$ and $B$, $D$ and $A$, and $C$ and $D$ are $F, \ G$, and $H$, respectively, then the quadrilateral $FEHG$ is a square. $$ $$ $$ $$…

Problem A convex quadrilateral $ABCD$ is a rhombus if $∠A=∠C, \ ∠B=∠D$ and $AB=BC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution The sum of the interior angles of a quadrilateral is $360°$. Thus, if $∠A = ∠C$ and…