Japan “Sangaku” Research Institute

Problem Let $D$ be the foot of the perpendicular drawn from the vertex $A$ to the opposite side $BC$ of $△ABC$, and let $E$ and $F$ be the midpoints of the sides $BC$ and $AB$, respectively. Then $$∠DFE=|∠B-∠C|$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Create an isosceles triangle $ALM$ that overlaps two sides $AB$ and $AC$ of a triangle $ABC$, extend $LM$ and $BC$, and let $N$ be their intersection. Then the straight line $LN$ intersects $AB$ at an angle equal to half the sum of its lower angles, and intersects the extension of the base of the…

Problem In $△ABC$, if the bisector of $∠A$ is $AN$ and the perpendicular line from $A$ to $BC$ is $AH$, then $$∠HAN=\frac{1}{2} |∠B-∠C|$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution When $AB>AC$,…

Problem In $△ABC$ where $AB>AC$, if we take $AD$ equal to $AC$ on $AB$, we have $$∠ADC=\frac{1}{2} (∠B+∠C) \quad and\quad ∠BCD=\frac{1}{2} (∠C-∠B)$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If $∠ADC=∠ACD$ and…

Problem When extending any side of a triangle, its exterior angle is greater than any of its inner opposite angles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution From the previous question $0053$,…