Japan “Sangaku” Research Institute

Problem If we take any point $F$ inside $△ABC$ and any two points $H$ and $I$ on $BC$, will the proposition $AB+AC>FH+FI$ always hold? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution  If…

Problem In $△ABC$, when $BC<AC<AB$, take any point $P$ in the triangle, create line segments $AP$, $BP$, and $CP$, and let $M$ be the intersection of $BC$ and the extension of $AP$, then, $$(1) \ AM+BC<AB+AC.$$ $$(2) \ AP+BP+CP<AB+AC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$…

Problem If we take an arbitrary point $O$ in $△ABC$ and represent $BC, \ CA$, and $AB$ by $a, \ b$, and $c$, respectively, we get $$a+b+c>AO+BO+CO>\frac{1}{2} (a+b+c).$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If we take any point $O$ within $△ABC$ and draw line segments $OB$ and $OC$, we get $$AB+AC>OB+OC, \qquad and \qquad ∠BOC>∠A.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If $BO$…

Problem If we take any points $D, \ E$, and $F$ on sides $BC, \ CA$, and $AB$ of $△ABC$, we get $$3(AD+BE+CF)<5(AB+BC+CA).$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution  If $AB$…

Problem When we take any two points $P$ and $Q$ inside or on the circumference of $△ABC$, the length of line segment $PQ$ does not exceed the length of the maximum side of this triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem In $△ABC$, when $AC$ is larger than $AB$, $AD$ connecting any $D$ on $BC$ and $A$ is smaller than $AC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If $AC>AB$, then from…

Problem The sum of two sides of a triangle is greater than the third side, and the difference between the two sides is less than the third side.   $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…