Question

Доказать, что если четырехугольник является одновременно вписанным и описанным, то его площадь равна корню из произведения сторон

Answer 1

По формуле Брахмагупты площадь вписанного в окружность четырехугольника равна:
$S = \sqrt{(p - a)(p - b)(p - c)(p - d)}$,
где a, b, c, d - стороны четырёхугольника, p - полупериметр.
Если в четырёхугольник можно вписать окружность, то суммы его противоположных сторон равны, т.е.
a + b = c + d
Уберём из формулы площади полупериметр, зная, что a + b = c + d:

$S = \sqrt{(\dfrac{1}{2}(a + b + c + d) - a)(\dfrac{1}{2}(a + b + c + d) - b)} \cdot \\ \\ \sqrt{\dfrac{1}{2}(a + b + c + d) - c)(\dfrac{1}{2}(a + b + c + d) - d)} = \\ \\ \sqrt{\bigg(\dfrac{1}{2}\bigg )^4(a + b + c -d)(a + b - c + d)(a - b+ c + d)(-a + b + c + d) } = \\ \\ \sqrt{\bigg(\dfrac{1}{2}\bigg )^4(c + d + c -d)(c + d - c + d)(a - b+ a + b)(-a + b + a + b) }= \\ \\ \sqrt{ \dfrac{1}{16} 2c \cdot 2d \cdot 2a \cdot 2b } = \sqrt{ \dfrac{1}{16}\cdot 16abcd } = \boxed{\sqrt{abcd} }$

Answer 2

Proof that if a quadrilateral is both inscribed and circumscribed, then its area is equal to the square root of the product of its sides

To prove this statement, let's consider a quadrilateral that is both inscribed and circumscribed.

Inscribed Quadrilateral: An inscribed quadrilateral is a quadrilateral whose vertices lie on a circle.

Circumscribed Quadrilateral: A circumscribed quadrilateral is a quadrilateral whose sides are tangent to a circle.

Let's denote the sides of the quadrilateral as a, b, c, and d, and the radius of the circumscribed circle as R.

To prove that the area of the quadrilateral is equal to the square root of the product of its sides, we will use the following steps:

1. The area of a quadrilateral can be calculated using Brahmagupta's formula, which states that the area (A) of a cyclic quadrilateral is given by the formula:

A = sqrt((s - a)(s - b)(s - c)(s - d)), where s is the semiperimeter of the quadrilateral.

2. In a circumscribed quadrilateral, the semiperimeter (s) is equal to the sum of the lengths of the sides divided by 2:

s = (a + b + c + d) / 2.

3. Substituting the value of s in the area formula, we get:

A = sqrt(((a + b + c + d) / 2 - a)((a + b + c + d) / 2 - b)((a + b + c + d) / 2 - c)((a + b + c + d) / 2 - d)).

4. Simplifying the expression, we have:

A = sqrt((s - a)(s - b)(s - c)(s - d)) = sqrt((a + b + c + d - a)(a + b + c + d - b)(a + b + c + d - c)(a + b + c + d - d)).

A = sqrt(abcd).

5. Therefore, the area of the quadrilateral is equal to the square root of the product of its sides, which is sqrt(abcd).

This proves that if a quadrilateral is both inscribed and circumscribed, then its area is equal to the square root of the product of its sides.