Problem Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$, and let $E$ and $F$ be the points where the bisectors of $∠ADB$ and $∠ADC$ intersect the sides $AB$ and $AC$, respectively. Then $$EF<BE+CF.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem In a triangle $ABC$, if we take any point $P$ on the bisector of $∠A$, we have $$|AB-AC|>|PB-PC|.$$ However, let $AB≠AC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution $(1)$ Let $AB>AC$. If we take a point $D$…

Problem First, create a new triangle $A’B’C’$ by connecting the intersections of the bisectors of each exterior angle of the original triangle $ABC$. Next, create a triangle $A^”B^”C^”$ by connecting the intersections of the bisectors of each exterior angle of the triangle $A’B’C’$. If you create new triangles one after another in this way, the…

Problem In a triangle $ABC$, if $∠B>∠C$, then the bisector $BD$ of $∠B$ is smaller than the bisector $CE$ of $∠C$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Since $∠B>∠C$, $$∠ABD>∠ACE.$$ Now, if we take $∠DBF$ equal to…

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, $$MN=MB+NC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$…