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…

Problem If the length of the long side of a right triangle is $a$ and the diameter of the circle inscribed in the three sides is $d$, what are the lengths of the short side $x$ and the hypotenuse $y$ respectively? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$…

Problem If the length of the short side of a right triangle is $a$ and the diameter of the circle inscribed in the three sides is $d$, what are the lengths of the long side $x$ and the hypotenuse $y$ respectively? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$…

Problem When the area of a right-angled triangle is $S$, and the diameter of a circle inscribed on the three sides is $d$, find the short side, long side, and hypotenuse of the right-angled triangle.　(In the figure below, $AB$ is the short side, $BC$ is the long side, and $CA$ is the hypotenuse.) $$ $$…

Problem If the length of the hypotenuse of a right-angled triangle is $a$ and the diameter of the circle inscribed in the right-angled triangle is $d$, what are the lengths of the short and long sides of the right-angled triangle?　(In the figure below, $AB$ is the short side and $BC$ is the long side.) $$…

Problem Find the diameter of a circle inscribed in three sides of a right triangle $ABC$ ($\angle B$ is a right angle). $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let the lengths…