Collinearity Proof Online Calculator
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= 1/2 |(x1*y2 + x2*y3 + x3*y1) - (x2*y1 + x3*y2 + x1*y3)|
= 1/2 |({{ x1 }}*{{ y2 }} + {{ x2 }}*{{ y3 }} + {{ x3 }}*{{ y1 }}) - ({{ x2 }}*{{ y1 }} + {{ x3 }}*{{ y2 }} + {{ x1 }}*{{ y3 }})|
= {{ areaCalculation }}
{{ result }}Collinearity in geometry is a fundamental concept that involves determining whether three points lie on the same straight line. This online calculator provides a straightforward way to verify the collinearity of three points by computing the area formed by these points. If the area is zero, the points are collinear; otherwise, they are not.
Historical Background
The concept of collinearity dates back to the early days of geometry, where the spatial relationships between points were essential for understanding shapes, lines, and angles. The methods for proving collinearity have evolved, from visual inspection and geometric constructions to algebraic and analytic techniques.
Calculation Formula
The collinearity of three points \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\) can be determined using the area of the triangle they form:
\[ \text{Area} = \frac{1}{2} |(x_1y_2 + x_2y_3 + x_3y_1) - (x_2y_1 + x_3y_2 + x_1y_3)| \]
If the area is \(0\), the points are collinear.
Example Calculation
Consider points \(A(1, 2)\), \(B(4, 5)\), and \(C(2, 3)\). The area of the triangle formed by these points is calculated as:
\[ \text{Area} = \frac{1}{2} |(1 \cdot 5 + 4 \cdot 3 + 2 \cdot 2) - (4 \cdot 2 + 2 \cdot 5 + 1 \cdot 3)| = 0 \]
Since the area is \(0\), points \(A\), \(B\), and \(C\) are collinear.
Importance and Usage Scenarios
Checking for collinearity is crucial in various fields such as computer graphics, robotics, and architectural design, where understanding the spatial relationships between points is necessary. It is also a key concept in mathematics and physics for solving problems related to vectors, forces, and motion.
Common FAQs
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What does collinear mean?
- Collinear means that three or more points lie on the same straight line.
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How can you tell if 3 points are collinear?
- If the area of the triangle formed by the three points is zero, they are collinear.
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Can this method be used for more than three points?
- For more than three points, you would check collinearity in pairs or use other algebraic methods to determine if they all lie on the same line.
This calculator simplifies the verification of collinearity, making it accessible to anyone interested in geometry, from students to professionals.