Japan “Sangaku” Research Institute

Problem In a rectangular triangle $ABC$ with $∠A$ as a right angle, if $D$ and $E$ are placed on the hypotenuse $BC$ such that $BD=AB$ and $CE=AC$, then $∠CAD$ and $∠BAE$ are equal to half of $∠B$ and $∠C$ respectively. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Draw a line parallel to the hypotenuse $BC$ from the vertex $A$ of the right isosceles triangle $ABC$, and place a point $D$ on it so that $BD=BC$. Let $E$ be the intersection point of $BD$ and $AC$. Then,$$CD=CE.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Draw a straight line $AD$ parallel to the side $BC$ from the vertex $A$ of a right triangle $ABC$ with $∠C=∠R$, and let $F$ be the intersection point of the sides $AC$ and $BD$. Then, if $FD=2AB$, $$∠ABF=2∠FBC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Let $D$ be the foot of the perpendicular line drawn from the right-angled vertex $A$ of a rectangular triangle $ABC$ to the hypotenuse $BC$. Then, the sum of the diameters of the inscribed circles of $△ABC, \ △ABD$ and $△ACD$ is equal to twice the length of $AD$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…

Problem In a right triangle $ABC$, $$∠A=∠R.$$Let $D$ be the foot of the perpendicular line drawn from $A$ to $BC$, $E$ be the point where the bisector of $∠B$ intersects $AC$, $F$ be the foot of the perpendicular line drawn from $E$ to $BC$, and $G$ be the point of intersection of $AD$ and $BE$.…

Problem Let $H$ be the foot of the perpendicular line drawn from the right-angled vertex $A$ of a triangle $ABC$ to $BC$. Inscribe squares, of which the two sides touch $AH$ and $BC$, in each of $△ABH$ and $△ACH$. Then, the sum of the length of one side of one square and that of the…

Problem Take a point $D$ on the extension of the side $AB$ of a triangle $ABC$ such that $AB=BD$, and let $E$ be the point closest to $B$ of the two that divide $BC$ into thirds. Then, if $AE=\frac{1}{3} CD$, $△ABC$ is a right triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$…

Problem If $D$ is the foot of the perpendicular line drawn from the right-angled vertex $A$ of a rectangular triangle $ABC$ to the hypotenuse $BC$, then$$AD+BC>AB+AC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Take a point $E$ on…

Problem If $D$ is the intersection point of the bisector of the right-angled vertex $A$ of a rectangular triangle $ABC$ and the line that passes through the midpoint $M$ of the hypotenuse $BC$ and is perpendicular to $BC$, then $$MA=MD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem If you drop a perpendicular line $AD$ from the right-angled vertex $A$ of a rectangular triangle $ABC$ to the side $BC$, $$∠C=∠BAD \qquad and \qquad ∠B=∠CAD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution $△ABC$ and $△DBA$ share…