Japan SANGAKU Research Institute

Problem The intersection points of the angle bisectors of a quadrilateral $ABCD$ with $∠A=∠C$ form an isosceles trapezoid. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution   As shown in the figure, let $G, \ H, \ K$, and…

Problem In trapezoid $ABCD$ with $AD∥BC$, let $P$ and $Q$ be the midpoints of the diagonals $BD$ and $AC$, respectively. Then the segment $PQ$ has length equal to one half of the difference between the two bases. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem The segment joining the midpoints of two opposite sides of a trapezoid is parallel to the bases and has length equal to half the sum of the bases. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $AB$…

Problem In a convex quadrilateral, if $∠B = ∠C$ and $AB = CD$, then the quadrilateral is an isosceles trapezoid. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $E$ be the point where the line through $A$ parallel…

Problem Given two bases $a$ and $b$ and two diagonals $m$ and $n$, prove that a trapezoid can be constructed if and only if $$m+n>a+b>|m−n|.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $E$ be the point where…