Use these sample points to verify your results quickly.
| Mode | Inputs | Key output |
|---|---|---|
| Circumcircle | A(1,2), B(7,3), C(4,8) | Center and radius, plus both bisectors |
| Segment bisector | P1(0,0), P2(6,4) | Bisector equation and midpoint |
1) Midpoint of a segment
The midpoint of two points (x₁, y₁) and (x₂, y₂) is M = ((x₁+x₂)/2, (y₁+y₂)/2).
2) Perpendicular bisector in general form
Let the segment direction be d = (dx, dy) = (x₂-x₁, y₂-y₁). A perpendicular bisector passes through M and satisfies:
dx·x + dy·y = dx·Mₓ + dy·Mᵧ
3) Circumcenter from two bisectors
For three non-collinear points, the circle center is the intersection of the bisectors of AB and BC. Solve the two linear equations to obtain (h, k).
4) Radius and circle equation
Radius r is the distance from (h, k) to any vertex, r = √((x-h)² + (y-k)²). The standard circle form is (x-h)² + (y-k)² = r².
- Select a mode: three points for a circle, or two points for a bisector.
- Enter coordinates using any consistent unit system.
- Choose decimal places for neat output formatting.
- Press Calculate to show results above the form.
- Use the download buttons to export a CSV or PDF report.
1) What this calculator solves
This tool connects perpendicular bisectors with circle geometry. In segment mode, it produces the exact line that cuts a segment in half at 90°. In three‑point mode, it builds a circumcircle by intersecting two bisectors, then reports the center and radius.
2) Input data and coordinate quality
Coordinates can be integers or decimals, in any units. The decimal‑places option controls rounding for display only, while the internal computation keeps full precision. If your points come from measurements, use more decimals to reduce rounding noise in slopes, intersections, and radius values.
3) Midpoint data you should verify
Each bisector is anchored at a midpoint. For a segment, the midpoint is ((x₁+x₂)/2, (y₁+y₂)/2). For a triangle, the calculator forms midpoints for AB and BC. Checking these midpoints against your sketch is a quick validation step before trusting the final circle.
4) Bisector equation formats and meaning
The perpendicular bisector is displayed in general form and an alternate form when possible. General form Ax + By = C works for all cases, including vertical and horizontal lines. If B ≠ 0, the alternate form becomes y = m x + b, making slope comparisons easier.
5) Circumcenter intersection data
The circumcenter (h, k) is found by solving the two linear equations from the bisectors of AB and BC. When points are collinear (or nearly collinear), the determinant becomes very small and a unique intersection does not exist. The calculator detects this and warns you.
6) Circle outputs you can reuse
Once the center is known, the radius is computed as the distance from (h, k) to any vertex. The page shows both standard form (x−h)²+(y−k)²=r² and an expanded form x²+y²+Dx+Ey+F=0. Expanded form is useful for algebra and coding.
7) Reporting, exports, and practical uses
Use the CSV download for spreadsheets and quick auditing, and use the PDF download for a clean submission or printout. Common uses include constructing circles through three survey points, verifying triangle properties, designing arcs in CAD workflows, and teaching coordinate geometry with consistent step notes.
1) What if my three points are in a straight line?
If the points are collinear, the perpendicular bisectors are parallel, so they never intersect at a single center. The calculator shows an error because no unique circumcircle exists for collinear points.
2) Which two bisectors does the tool use for the circle?
It uses the perpendicular bisectors of AB and BC. Any pair of side bisectors would work for a non‑collinear triangle, but using adjacent sides provides stable equations and a clear geometric interpretation.
3) Why do I see only a general-form line sometimes?
If the bisector is vertical or nearly vertical, slope form is undefined or unstable. General form Ax + By = C remains valid in every case, so it is always shown as the reliable representation.
4) Does changing decimal places change the real result?
No. Decimal places control formatting only. Calculations run using full numeric precision first, then the output is rounded for display. For careful work, increase decimal places to see more digits.
5) Can I use this for real measurements in meters or inches?
Yes. The math is unit‑consistent: the center uses the same units as your coordinates, and the radius is reported in those units as well. Just keep one unit system throughout your inputs.
6) What do the CSV and PDF exports include?
Exports include the chosen mode, an input summary, circle center and radius when available, and the main bisector equations. It’s designed for quick sharing, record‑keeping, and attaching results to assignments.