How can I determine the center of a circle given three points on its circumference?

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Finding the center of a circle given three points on its circumference involves a bit of geometry and algebra, but it can be done through a systematic approach. Here’s how you can do it step by step:

  1. Identify the Points: Let’s denote the three points as A(x1, y1), B(x2, y2), and C(x3, y3).
  2. Calculate the Midpoints: First, calculate the midpoints of two line segments connecting these points. For example, find the midpoint of segment AB and segment BC:
    • Midpoint of AB: MAB = ((x1 + x2) / 2, (y1 + y2) / 2)
    • Midpoint of BC: MBC = ((x2 + x3) / 2, (y2 + y3) / 2)
  3. Find the Slopes of the Lines: Next, calculate the slopes of lines AB and BC:
    • Slope of AB: slopeAB = (y2 – y1) / (x2 – x1)
    • Slope of BC: slopeBC = (y3 – y2) / (x3 – x2)
  4. Calculate the Perpendicular Slopes: The center of the circle lies along the perpendicular bisectors of the chords. Calculate the slopes of these perpendicular bisectors:
    • Perpendicular slope to AB: perpendicular slopeAB = -1 / slopeAB
    • Perpendicular slope to BC: perpendicular slopeBC = -1 / slopeBC
  5. Write the Equations of the Perpendicular Bisectors: Using the midpoints and the slopes, you can write the equations of the perpendicular bisectors:
    • Equation of perpendicular bisector of AB: y – MAB.y = perpendicular slopeAB * (x – MAB.x)
    • Equation of perpendicular bisector of BC: y – MBC.y = perpendicular slopeBC * (x – MBC.x)
  6. Solve the Simultaneous Equations: Now, you have two equations (one from each perpendicular bisector). Solve these simultaneously to find the coordinates (h, k) of the circle’s center.
  7. Verification: Once you have the center (h, k), you can verify that the distances from this point to all three original points A, B, and C are equal, confirming that they do indeed lie on the circumference of the circle.

This method combines basic principles of geometry and algebra, making it accessible and straightforward. Whether you’re solving a geometry problem, working on a design project, or just curious, this technique can help you accurately find the center of a circle defined by three points.

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