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…

Problem The perimeter of a quadrilateral is greater than the sum of its diagonals but less than twice the sum. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△ABC$ and $△ACD$, $$AC<AB+BC,$$ $$AC<CD+DA,$$ $$BD<BC+CD,$$ $$BD<AB+DA,$$ $$∴ \ 2(AC+BD)<2(AB+BC+CD+DA),$$…

Problem The sum of the interior angles of a quadrilateral is equal to four right angles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution As shown in the diagram above, if you draw a diagonal $AC$ on a quadrilateral…