Japan “Sangaku” Research Institute

Problem Let $Q$ and $R$ be the points where the perpendicular line passing through any point $P$ on the base $BC$ of an isosceles triangle $ABC$ with vertex $A$ intersects with the sides $AB$ and $AC$ (or their extensions), respectively. Then, if $PQ+PR$ is always equal to the side $BC$, $$∠A=∠R.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$…

Problem If the sum of perpendicular lines $PD$ and $PF$ drawn from any point $P$ on one side of an isosceles triangle $ABC$ with vertex $A$ to the other two sides is equal to the height $AH$, then $△ABC$ is an equilateral triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Extend the side $AB$ of an isosceles triangle $ABC$ with $A$ as the vertex, take a point $D$ such that $AB=BD$, and let $E$ be the midpoint of $AB$, then $$CD=2CE.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If two triangles $ABC$ and $DBC$ have a common base $BC$, and $AD$ is parallel to $BC$, and $△ABC$ is an isosceles triangle, then the perimeter of $△ABC$ is less than that of $△DBC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

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…