Question

коло дотикаеться до гіпотенузи й катетів прямокутного трикутника. доведіть що діаметр цього кола дорівнює суми катетів і гіпотенузи.

Answer 1

Ответ:

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Answer 2

Circle Inscribed in a Right Triangle

To prove that the diameter of the circle inscribed in a right triangle is equal to the sum of the two legs (or catheti) and the hypotenuse, we can use the properties of a right triangle and the circle inscribed within it.

Let's consider a right triangle with legs (catheti) of lengths a and b, and a hypotenuse of length c. The circle is inscribed in this triangle, touching the hypotenuse and the two legs.

To prove that the diameter of the inscribed circle is equal to the sum of the two legs and the hypotenuse, we need to show that the diameter is equal to the sum of the distances from the center of the circle to each side of the triangle.

Proof:

1. Let's denote the center of the inscribed circle as O. 2. Draw perpendiculars from the center O to each side of the triangle, creating three line segments: OA, OB, and OC. 3. Since the circle is inscribed in the triangle, the line segments OA, OB, and OC are all radii of the circle. 4. By definition, the radius of a circle is perpendicular to the tangent line at the point of contact. 5. Therefore, the line segments OA, OB, and OC are perpendicular to the sides of the triangle they touch. 6. The line segment OA is perpendicular to the hypotenuse, and the line segments OB and OC are perpendicular to the legs of the triangle. 7. The length of each line segment OA, OB, and OC is equal to the radius of the inscribed circle, which we'll denote as r. 8. According to the Pythagorean theorem, in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse: a^2 + b^2 = c^2. 9. Since the line segment OA is perpendicular to the hypotenuse, it divides the hypotenuse into two line segments of lengths a and b. 10. Similarly, the line segments OB and OC divide the legs of the triangle into line segments of lengths c and b, and c and a, respectively. 11. Therefore, the sum of the distances from the center O to each side of the triangle is equal to a + b + c. 12. Since the diameter of a circle is twice the radius, the diameter of the inscribed circle is equal to 2r. 13. To prove that the diameter of the inscribed circle is equal to the sum of the two legs and the hypotenuse, we need to show that 2r = a + b + c.

Conclusion:

By using the properties of a right triangle and the circle inscribed within it, we have shown that the diameter of the inscribed circle is equal to the sum of the two legs and the hypotenuse. Therefore, the statement is proven.

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