Japan “Sangaku” Research Institute

P roblem In a triangle $ABC$, if the intersection of the bisectors of $∠B$ and $∠C$ is $O$, then $$∠BOC=90°+\frac{1}{2}∠A.$$ Also, if the intersection of the bisectors of the exterior angles at $B$ and $C$ is $O’$, then $$∠BO’ C=90°-\frac{1}{2}∠A.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

P roblem Is the statement “The bisector of the apex angle of a triangle and the perpendicular bisector of the opposite side intersect at a point outside the triangle” correct? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution The…

P roblem In a triangle $ABC$, the angle formed by the intersection $E$ of the bisectors of $∠B$ and the exterior angle of $∠C$ is equal to half of $∠A$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let…

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…