Japan SANGAKU Research Institute

Problem Consider the planar figure shown in the diagram, composed of a sequence of ten line segments. What is the sum of the angles $A, \ B, \ …, \ J$ formed by each pair of adjacent line segments in this figure? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem The exterior angle at each vertex of a regular $n$-gon is $$\frac{4}{n} ∠R.$$ The interior angle at each vertex is $$\left( 2−\frac{4}{n} \right)∠R.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Drawing $n−3$ diagonals from a single vertex of…

Problem When a hexagon is inscribed in a circle, the sum of the interior angles at alternating vertices is equal. Does the converse also hold? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution As shown in the figure, drawing…

Problem State the necessary and sufficient condition under which a polygon can be inscribed in a circle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution The necessary and sufficient condition for a polygon to be inscribed in a circle…

Problem In a convex polygon, there can be at most three acute interior angles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution As shown in the figure, when the interior angle at a vertex is acute, the corresponding exterior…

Problem Find the number of diagonals in a convex $n$‑gon. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Since each vertex is connected by a diagonal to every other vertex except its two adjacent neighbors, the number of diagonals…

Problem The sum of the interior angles of a convex $n$-gon is $(2n−4)∠R$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If an arbitrary point $O$ is chosen inside this convex polygon, the polygon can be divided into $n$…