given a convex quadrilateral ABCD circumscribed about a circle of diameter 1 .Inside ABCD,there is a point M such that |MA|^2+|MB|^2+|MC|^2+|MD|^2=2. Find the area of ABCD

To solve the problem of finding the area of the convex quadrilateral ABCD that is circumscribed about a circle of diameter 1, we can leverage some geometric properties and relationships involving points and distances. The key information provided is that the sum of the squares of the distances from an internal point M to the vertices of the quadrilateral equals 2. Let's break this down step by step.

Understanding the Geometry

First, since ABCD is circumscribed about a circle, it means that the circle is tangent to all four sides of the quadrilateral. The radius of the circle, given that the diameter is 1, is therefore 0.5. This circle is known as the incircle of the quadrilateral.

Using the Distance Formula

The condition given is |MA|² + |MB|² + |MC|² + |MD|² = 2. We can use a well-known result in geometry that relates the area of a quadrilateral circumscribed about a circle to the distances from any internal point to its vertices. Specifically, for a quadrilateral ABCD with an incircle, the area can be expressed using the formula:

• Area = r * s

Here, r is the radius of the incircle, and s is the semiperimeter of the quadrilateral. However, we can also use the relationship involving the distances from point M.

Applying the Distance Condition

For any point M inside a quadrilateral ABCD, the following relationship holds:

• |MA|² + |MB|² + |MC|² + |MD|² = 4R² + 4d²

Where R is the circumradius of the quadrilateral and d is the distance from M to the center of the incircle. In our case, we know that |MA|² + |MB|² + |MC|² + |MD|² = 2. Since the radius of the incircle r is 0.5, we can substitute R = 0.5 into the equation:

Calculating the Area

Given that the radius r = 0.5, we can find the area of the quadrilateral using the relationship:

• Area = r² * (|MA|² + |MB|² + |MC|² + |MD|²) / 4

Substituting the known values:

• Area = (0.5)² * (2) / 4
• Area = 0.25 * 2 / 4
• Area = 0.25 / 2
• Area = 0.125

Final Result

Thus, the area of the convex quadrilateral ABCD is 0.125 square units. This result illustrates how the geometric properties of circumscribed quadrilaterals and the distances from internal points can be effectively used to derive important measurements.