Japan “Sangaku” Research Institute

Problem Take the points $E$ and $F$ on the sides $AB$ and $AC$ of $△ABC$ respectively, and $BE$ and $CF$ intersect at $G$. If $2GE=GB$ and $2GF=GC$,  then $G$ is the center of gravity of $△ABC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem In $△ABC$, let the two medians be $BE$ and $CF$.   If $AB>AC$, then $$BE>CF.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let the other median be $AD$, and the center of gravity of $△ABC$ be $G$. $△ABD$…

Problem The medians of the three sides of a triangle intersect at a point located $\frac{2}{3}$ of the way from each vertex. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let the three medians of $△ABC$ be $AM, \…

Problem In $△ABC$, let $D$ be the midpoint of $AB$. Then, the length of the line segment connecting $D$ and the midpoint $E$ of $AC$ is $\frac{1}{2} BC$. State the converse of this theorem and determine the conditions for it to hold. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Let $D$ be the midpoint of the side $AB$ of $△ABC$. If a circle with radius $\frac{1}{2} BC$ centered on $D$ is drawn, prove that it intersects with $AC$ at one or two points, and determine the conditions for there to be only one point of intersection. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…

Problem Is the following statement correct ?   Let the line passing through the midpoint $M$ of the side $AB$ of $△ABC$ intersect with the point $N$ on the side $AC$. In this case, if $MN=\frac{1}{2} BC$, then $$MN∥BC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem In $△ABC$, suppose $AB>AC$. Then, if we draw the median line $AD$, $$∠BAD<∠CAD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $E$ be the intersection point of the line parallel to $AC$ that passes through $B$ and…

Problem Let $AD, \ BE$, and $CF$ be the three medians of $△ABC$, and $G$ be their intersection point. Then, $$AD+BE+CF<AB+BC+CA<2 \ (AG+BG+CG).$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution First, from the problem $0098$, $$AD+BE+CF<AB+BC+CA. \qquad [1]$$…

Problem Let $AD, \ BE$ and $CF$ be the three median lines of $△ABC$, respectively. Then, $$\frac{1}{2} (AB+BC+CA)<AD+BE+CF<AB+BC+CA.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution From the problem $0097 \ (2)$, $$\frac{1}{2} (AB+CA-BC)<AD, \qquad \frac{1}{2} (AB+BC-CA)<BE,$$ $$and \qquad…