Question

Довести, що якщо діагоналі вписаного в коло чотирикутника взаємно перпендикулярні, то сума квадратів протилежних сторін чотирикутника дорівнює квадрату діаметра цього кола

Answer 1

Пусть ABCD - наш четырехугольник.
sin(∠DAC)=sin(90°-∠ADB)=cos(∠ADB)=cos(∠ACB) т.к. диагонали перпендикулярны и углы ADB и АСВ равны как вписанные.

По теореме синусов 2R=AB/sin(∠ACB) и 2R=DC/sin(∠DAC), откуда
AB=2R·sin(∠ACB), DC=2R·sin(∠DAC)=2R·cos(∠ACB). Значит
AB²+DC²=4R²(sin²(∠ACB)+cos²(∠ACB))=4R².

Answer 2

The Relationship Between the Diagonals and Sides of an Inscribed Quadrilateral

To prove that if the diagonals of an inscribed quadrilateral are mutually perpendicular, then the sum of the squares of the opposite sides of the quadrilateral is equal to the square of the diameter of the circle, we can use the following steps:

1. Let ABCD be the inscribed quadrilateral with diagonals AC and BD, and let O be the center of the circle.

2. Since the diagonals AC and BD are mutually perpendicular, we can conclude that triangles AOB and COD are right triangles.

3. Using the properties of right triangles, we can express the lengths of the sides of the quadrilateral in terms of the radius of the circle and the angles formed by the diagonals.

4. Let r be the radius of the circle. From triangle AOB, we have: - AB = 2r * sin(AOB) - OA = OB = r

5. Similarly, from triangle COD, we have: - CD = 2r * sin(COD) - OC = OD = r

6. Now, let's calculate the sum of the squares of the opposite sides of the quadrilateral: - AB^2 + CD^2 = (2r * sin(AOB))^2 + (2r * sin(COD))^2 - AB^2 + CD^2 = 4r^2 * sin^2(AOB) + 4r^2 * sin^2(COD) - AB^2 + CD^2 = 4r^2 * (sin^2(AOB) + sin^2(COD))

7. Since AOB and COD are right angles, we know that sin(AOB) = sin(COD) = 1.

8. Substituting the values, we