Problem In a triangle $ABC$, let $O$ be the intersection of the bisectors of $∠B$ and $∠C$, and let $M$ and $N$ be the intersections of the straight line passing through $O$ and parallel to $BC$, and $AB$ and $AC$, respectively. Then, $$MN=|MB-NC|.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$…

Problem Let $O$ be the intersection of the bisectors of $∠B$ and $∠C$ of a triangle $ABC$, and let $M$ and $N$ be the intersections of $AB$ and $AC$ with a straight line drawn through $O$ parallel to $BC$, respectively. Then, $$MN=MB+NC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$…

Problem Let $O$ be the intersection of the bisectors of $∠B$ and $∠C$ of a triangle $ABC$, and let $D$ be the intersection of the extension of $AO$ and $BC$. If the perpendicular from $O$ to $BC$ is $OE$, then $$∠BOE=∠COD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$…

Problem If $D$ is the point where the bisector of $∠A$ intersects the side $BC$ in the triangle $ABC$, then $$AB>BD \qquad and \qquad AC>CD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…