Center of Circle Calculator
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The Center of Circle Calculator is designed to determine the center of a circle given three points that lie on its circumference. This is an important tool in geometry and mathematics, especially for those studying circles or needing to solve problems involving circumcircles.
Historical Background
The problem of determining the center of a circle given points on its circumference has roots in classical geometry. Ancient Greek mathematicians, such as Euclid, explored the properties of circles, including finding their centers based on various given elements. Understanding the relationship between different points on a circle was fundamental to early advancements in geometry.
Calculation Formula
The general approach to finding the center of a circle passing through three non-collinear points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is based on solving the perpendicular bisectors of chords formed by these points. The formulas are derived using determinants and algebra:
\[ a = x_1 \times (y_2 - y_3) + x_2 \times (y_3 - y_1) + x_3 \times (y_1 - y_2) \]
\[ X = \frac{(x_1^2 + y_1^2) \times (y_2 - y_3) + (x_2^2 + y_2^2) \times (y_3 - y_1) + (x_3^2 + y_3^2) \times (y_1 - y_2)}{2a} \]
\[ Y = \frac{(x_1^2 + y_1^2) \times (x_3 - x_2) + (x_2^2 + y_2^2) \times (x_1 - x_3) + (x_3^2 + y_3^2) \times (x_2 - x_1)}{2a} \]
Example Calculation
Suppose you are given three points: (1, 2), (4, 6), and (5, 3). By substituting these values into the formula:
- \(a = 1 \times (6 - 3) + 4 \times (3 - 2) + 5 \times (2 - 6) = -9\)
- \(X = \frac{(1^2 + 2^2) \times (6 - 3) + (4^2 + 6^2) \times (3 - 2) + (5^2 + 3^2) \times (2 - 6)}{2a} = 3\)
- \(Y = \frac{(1^2 + 2^2) \times (5 - 4) + (4^2 + 6^2) \times (1 - 5) + (5^2 + 3^2) \times (4 - 1)}{2a} = 2\)
Thus, the center of the circle is at (3, 2).