Enter three triangle vertices to calculate side midpoints, perpendicular bisector equations, circumcenter position, circumradius, and a clear coordinate graph.
Calculator Inputs
Use coordinate values for points A, B, and C.
Plotly Graph
The graph plots the triangle, three side bisectors, the circumcenter, and the circumcircle.
Example Data Table
| Example | Coordinates | Circumcenter | Radius | AB Bisector |
|---|---|---|---|---|
| Example 1 | A(0,0), B(6,0), C(2,6) | (3.0000, 2.3333) | 3.8006 | x = 3.0000 |
| Example 2 | A(-2,1), B(4,3), C(1,7) | (0.5000, 3.5000) | 3.5355 | y = -3.0000x + 8.0000 |
| Example 3 | A(0,0), B(4,0), C(0,3) | (2.0000, 1.5000) | 2.5000 | x = 2.0000 |
Formula Used
1. Midpoint of a side: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
2. Perpendicular bisector condition: the bisector passes through the side midpoint and stays perpendicular to the side.
3. Standard bisector equation for side endpoints P₁ and P₂: (x₂ - x₁)x + (y₂ - y₁)y - ((x₂ - x₁)Mₓ + (y₂ - y₁)Mᵧ) = 0
4. Circumcenter: the common intersection point of the three perpendicular bisectors.
5. Circumradius: distance from the circumcenter to any vertex, using r = √((x - h)² + (y - k)²).
How to Use This Calculator
- Enter the X and Y coordinates for vertices A, B, and C.
- Choose the decimal precision for the final output.
- Press Calculate Perpendicular Bisectors.
- Review the midpoints, slopes, line equations, circumcenter, and circumradius.
- Use the chart to inspect the geometry visually.
- Export the computed results as CSV or PDF when needed.
Frequently Asked Questions
1. What does a triangle perpendicular bisector do?
It cuts a triangle side into two equal parts and meets that side at a right angle. Every point on the bisector stays equally distant from the side’s two endpoints.
2. Why do all three perpendicular bisectors meet together?
They intersect at the circumcenter. That point is equally distant from all three vertices, so it becomes the center of the circumcircle passing through the triangle’s corners.
3. What happens if my three points are collinear?
A valid triangle does not exist. The calculator stops because circumcenter and circumcircle formulas require a nonzero triangle area.
4. Can this calculator handle decimal coordinates?
Yes. You can enter integers or decimals for every coordinate. The output precision field lets you control how many decimal places appear in the results.
5. Does the circumcenter always lie inside the triangle?
No. It lies inside acute triangles, on the midpoint of the hypotenuse for right triangles, and outside obtuse triangles.
6. What is the difference between a bisector midpoint and circumcenter?
A midpoint belongs to one side only. The circumcenter is the shared intersection point of all perpendicular bisectors and defines the circumcircle center.
7. Why does the graph show a circle too?
The circle confirms the circumcenter visually. Each vertex should lie on that circle, proving the center is equally distant from all triangle corners.
8. When is this calculator useful?
It helps in coordinate geometry homework, drafting layouts, triangle proofs, and any problem requiring midpoint analysis, circumcenter location, or perpendicular bisector equations.