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.   $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If two sides of one triangle are equal to two sides of another triangle, but the third sides are unequal, the angle opposite the larger side is greater than the angle opposite the smaller side. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem When two sides of one triangle are equal to two sides of another triangle, and the angles between the two sides are unequal, the side facing the larger angle is greater than the side facing the smaller angle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$…

Problem When the two angles of a triangle are unequal, the side opposite the larger angle is longer than the side opposite the smaller angle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ SolutionIn…