Japan “Sangaku” Research Institute

Problem If four equal circles are contained in a right triangle whose short side is $3 ins.$ and long side is $4 ins.$, as shown in the figure, find the diameter d of the circles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem When two equal circles are placed inside a right triangle with a length of $3 \ ins.$ and a width of $4 \ ins.$ as shown in the figure, find the diameter of the equal circles.   $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem As shown in the figure, when a large circle with a diameter of $450 \ cm$ and a small circle with a diameter of $288 \ cm$ are inscribed within a right triangle, find the lengths of the three sides of the right triangle.   $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$…

Problem When six spheres with a diameter of $8 \ cm$ are arranged on a flat surface as shown in the figure, and a sphere with a diameter of $10 \ cm$ is placed on top of them, find the height of this object. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$…

Problem If three balls with a diameter of $8 \ cm$ are arranged on a flat surface so that each ball touches the other two as shown in the figure, and a ball with a diameter of $7 \ cm$ is placed on top of them, find the height of the object. $$ $$ $$…

Problem As shown in the figure, parts of the long side and the hypotenuse of a right triangle are used as the two sides of a rhombus, and after cutting out the rhombus, each of the two remaining right triangles contains a circle inscribed on its three sides. Assume that the diameter of the upper…

Problem There is an isosceles trapezoid with a bottom side of $18 \ cm$, lateral sides of $32.8 \ cm$, and a top side of $3.6 \ cm$. Find the diameter of the circle inscribed in the top and lateral sides as shown. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$…

Problem If there is a square and a circle inscribed in a right triangle, and one side of the square is $l$ and the diameter of the circle is $d$, what is the sum of the three sides of the right triangle? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$…

Problem   If the diameter of the circle inscribed in the three sides of a right-angled triangle $ABC$ is $d$, and the length of the perpendicular $BD$ from the right angle $∠B$ to the hypotenuse $CA$ is known to be $a$, What are the lengths of the short side $AB$, the long side $BC$, and…

Problem If the differences between the hypotenuse $CA$ and the short side $AB$ of a right-angled triangle and between the hypotenuse $CA$ and the long side $BC$ are $a$ and $b$ respectively, what are the lengths of the short side, the long side, and the hypotenuse, and the diameter $d$ of the circle inscribed in…