Japan “Sangaku” Research Institute

Problem In an acute triangle $ABC$, let $∠B=2∠C$ and $AH⊥BC$. When $AB$ is extended to $D$ so that $BD=BH$, the extension of $DH$ passes through the midpoint $M$ of $AC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If we drop a perpendicular line $AB$ to the straight line $XY$ from a point $A$ that is not on $XY$, and draw hypotenuses $AC$, $AD$ and $AE$ on the same side as $AB$ so that $∠BAC$, $∠CAD$ and $∠DAE$ are equal, we have $$CB<DC<ED.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…

Problem In a triangle $ABC$, if $AB>AC$, and we take any point $P$ on the perpendicular $AD$ drawn from the vertex $A$ to the opposite side $BC$, then we have $$PB-PC>AB-AC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Suppose that any straight line passes through the vertex $A$ of a triangle $ABC$. The feet $D$ and $E$ of the perpendicular lines drawn from $B$ and $C$ to the above line are equidistant from the midpoint $M$ of the side $BC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$…

Problem If the feet of the perpendicular lines drawn from the two vertices $B$ and $C$ of a triangle $ABC$ to the opposite sides $AC$ and $AB$ are $E$ and $F$, respectively, then the straight line connecting the midpoint of the line segment $EF$ and the midpoint of the side $BC$ is perpendicular to $EF$.…

Problem The length of the bisector of one angle of a triangle is less than the average length of the two sides forming that angle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…

Problem The bisector $AE$ of the apex angle $∠A$ of a triangle $ABC$ lies between the median line $AM$ and the perpendicular $AD$ drawn from this apex angle to the opposite side. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem In $△ABC$, suppose $AC>AB$, and if we draw the perpendicular $AD$ from $A$ to $BC$, we have $$∠DAC>∠DAB \qquad and \qquad DC>DB$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Since $AD⊥BC$,…

Problem The sum of the lengths of perpendiculars drawn from the three vertices of a triangle to their opposite sides is less than the sum of the lengths of the three sides of the triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…