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…

Problem When the bisectors of the four angles of a quadrilateral pass through the same point, the sum of the lengths of one pair of opposite sides of the quadrilateral is equal to the sum of the lengths of the other pair of opposite sides. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$…