Sangaku Quest

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. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$…

Problem If the longest side of a quadrilateral $ABCD$ is $AD$ and the shortest side is $BC$, then $∠BCD$ is greater than $∠BAD$ and $∠ABC$ is greater than $∠ADC$: $$∠BCD>∠BAD \qquad and \qquad ∠ABC>∠ADC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Let $ABCD$ be any quadrilateral, and let there be an interior point $O$ which is not the intersection of the diagonals; then the sum of $OA, \ OB, \ OC$ and $OD$ is greater than the sum of both diagonals $AC$ and $BD$: $$OA+OB+OC+OD＞AC+BD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$…

Problem In the following $(1)$ to $(4)$, for the condition on the left about a quadrilateral, if the condition on the right is a necessary and sufficient condition, answer $A$; if it is a necessary but not sufficient condition, answer $B$; if it is a sufficient but not necessary condition, answer $C$; if it is…