Japan “Sangaku” Research Institute

Problem When the bisectors of each angle of a quadrilateral intersect to form a second quadrilateral, $(1)$ the sum of the two opposing angles of the quadrilateral is two right angles; $(2)$ if the original quadrilateral is a parallelogram, the second quadrilateral is a rectangle, and its two diagonals are parallel to each side of…

Problem In a quadrilateral $ABCD$, if the angles of intersection of the bisectors of $∠A$ and $∠B$, and $∠A$ and $∠C$ are $α$ and $β$, respectively, then $$α=\frac{1}{2} (∠C+∠D) \qquad and \qquad β=\frac{1}{2} |∠B-∠D|.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem In a quadrilateral $ABCD$ such that $AB=CD$, if the midpoints of $DA$ and $BC$ are $P$ and $Q$ respectively, and the midpoints of the diagonals $AC$ and $BD$ are $M$ and $N$ respectively, then $PQ$ and $MN$ are perpendicular to each other. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem The length of the line segment connecting the midpoints $P$ and $Q$ of the opposing sides $AB$ and $CD$ of a quadrilateral $ABCD$ is not greater than half the sum of the lengths of the other two sides $BC$ and $DA$. If this line segment $PQ$ is equal to the sum of the lengths…

Problem If the lengths of the two sides $AB$ and $CD$ of a quadrilateral $ABCD$ are equal, then the extensions of these sides and the line connecting the midpoints $M$ and $N$ of the other two sides $AD$ and $BC$ will make equal angles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem The line segment joining the midpoints $M$ and $N$ of the two diagonals $BD$ and $AC$ of a quadrilateral $ABCD$ is not less than half the difference between the two opposite sides. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem By connecting the midpoints of each side of any quadrilateral $ABCD$ in turn, a parallelogram is created, of which the perimeter is equal to the sum of the diagonals $AC$ and $DB$ of the original quadrilateral. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem In a quadrilateral $ABCD$, when opposite $∠A$ and $∠C$ are equal, the bisectors of another pair of opposite $∠B$ and $∠D$ are parallel to each other. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $BM$ and $DN$…

Problem For a quadrilateral $ABCD$, if the bisectors of $∠A$ and $∠C$ intersect on the diagonal $BD$, then the bisectors of $∠B$ and $∠D$ intersect on the diagonal $AC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If the…