Problem In $△ABC$, extend the median $AM$ so that $MD=BC$. If $∠AMC=60°$, then $$BD⊥BC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Taking the midpoint $E$ of $MD$, $$ME=MB \qquad and \qquad ∠BME=60°.$$ Thus, $△MBE$ is an equilateral triangle,…

Problem In $△ABC$, let $AB>AC$, and take any point $P$ on the median line $AD$. Then, $$AB-AC>PB-PC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If we take a point $E$ on $AB$ so that $AC=AE$, $$AB-AC=AB-AE=EB.$$ Since $AB>AC$…

Problem Two lines $x$ and $y$ intersect at point $O$. On each of these lines are equal-length segments $AB$ and $CD$, and the midpoints of $AC$ and $BD$ are $M$ and $N$, respectively. Then, the line $MN$ has a constant direction regardless of the positions of $AB$ and $CD$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…

Problem Construct the median $AD$ of $△ABC$, and draw a line parallel to $AD$ from any point $P$ on the side $BC$. Let $Q$ and $R$ be the points where the line intersects with $AB$ and $AC$, or their extensions. Then, the length of $PQ+PR$ is constant regardless of the location of $P$. $$ $$ $$ $$ $\downarrow$…

Problem Let $D$ be the midpoint of the side $AB$ of $△ABC$, and the point $E$ be on the side $AC$ so that $AE∶EC=2∶1$. Moreover, let $O$ be the intersection of $CD$ and $BE$. Then, $$BE∶OE=4∶1.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If the medians $BE$ and $CF$ of $△ABC$ are extended so that $BE=EG$ and $CF=FH$, then $G, \ A$ and $H$ are collinear. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△AFH$ and $△BFC$, $$∠AFH=∠BFC, \qquad AF=BF,…

Problem The points $B$ and $C$ are on the same side of the line $XY$, and $A$ is on the opposite side. When the sum of the distances from $B$ and $C$ to $XY$ is equal to the distance from $A$ to $XY$,   $XY$ passes through the center of gravity of $△ABC$. $$ $$ $$ $$ $\downarrow$…

Problem If the feet of perpendicular lines drawn from vertices $A, \ B$, and $C$ to any line $XY$ passing through the center of gravity $G$ of $△ABC$ are $P, \ Q$, and $R$ respectively, then $$AP=BQ+CR.$$ However, $A$ is on the opposite side of $XY$ from $B$ and $C$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$…

Problem The sum of the lengths of the perpendicular lines $AL, \ BM$, and $CN$ drawn from the vertices of $△ABC$ to a line outside the triangle is equal to three times the length of the perpendicular line $GP$ drawn from the center of gravity $G$ of the triangle to the line outside the triangle.…