Japan SANGAKU Research Institute

Problem A straight line cannot intersect a circle at more than two points. Therefore, no three points on the circumference lie on a single straight line. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Suppose a straight line intersects…

Problem Prove each of the following statements: The perimeter of a square inscribed in a circle is less than the circumference of the circle. The circumference of a circle inscribed in a square is less than the perimeter of the square. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Every circle that passes through two fixed points $A$ and $B$ has its center on the perpendicular bisector of segment $AB$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Draw a line $XY$ through the midpoint $P$ of…

Problem Any diameter of a circle divides the circle into two congruent halves. In addition, every diameter serves as an axis of symmetry. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $AB$ be the diameter of circle $O$.…

Problem Among all circles that contain triangle $ABC$, which one has the smallest radius? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution In the case of an obtuse triangle (where $∠A$ is obtuse in the diagram), the smallest circle…

Problem A circle has only one center. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $O$ be one center of the circle and $O′$ be another. If $A$ and $B$ are the points where the line $OO′$ meets…