Problem Let $M$ be the midpoint of the base $BC$ of an isosceles triangle $ABC$.When a line passing through $M$ intersects with the side $AB$ at the point $D$ and with the extension of the side $AC$ at the point $E$,$$AB+AC<AD+AE.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem The line segment $BD$ joining the point $B$ of the base of an isosceles triangle $ABC$ with vertex $A$ to any point $D$ on $AC$ is greater than the line segment $DE$ joining the point $D$ to any point $E$ on $BC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem The two perpendicular lines drawn from the vertex of an isosceles triangle to the bisectors of the base angles are equal in length. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $D$ and $E$ be the feet…

Problem Let the vertex angle $A$ of an isosceles triangle $ABC$ be $120°$. Let $D$ be the foot of the perpendicular line drawn from the vertex angle $A$ to $BC$, and take any point $P$ on $AD$ and connect it to $B$ and $C$. The following inequality holds: $$AP+BP+CP>AB+AC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…

Problem If the bisectors of $∠B$ and $∠C$ of a triangle $ABC$ are equal, then the triangle is isosceles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If we assume that $∠B>∠C$, then from the problem $0080$,$$BD<CE.$$Similarly, if we…

Problem The straight line connecting the intersection of perpendicular lines drawn from both ends of the base of an isosceles triangle to the vertex bisects the apex angle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If the intersection…

Problem The medians drawn from both ends of the base of an isosceles triangle to the opposite sides are equal. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution $△DBC$ and $△ECB$ share the side $BC \ (=CB)$, $$DB=EC \qquad…

Problem If the bisector of the vertex angle $A$ of a triangle $ABC$ is perpendicular to the base $BC$, then the triangle is an isosceles triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If the intersection point of…

Problem In a triangle $ABC$, if $AB=AC$, then $$∠B=∠C.$$ And vice versa. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If we turn $△ABC$ over and make it $△AC’ B’$, $$AB=AC’ \qquad and \qquad AC=AB’, \qquad (∵ \ AB=AC)$$…