Testing whether four points really share one circle
Four dots are marked on a page. Do all four sit on one circle?
- Three points, fixing a circle and its centre.
- A fourth point, exactly on that circle.
- A different fourth point, sitting off the circle.
The test
Only if the fourth is on the circle through the other three.
Points on one circle are called concyclic. Three points not in a line are always concyclic. Pick three of the four points that are not in a line. One circle passes through them. The worked example below finds its centre. Then measure the fourth point's distance from that centre. The four points are concyclic only when that distance equals the radius.
A second test, with angles
Join the four points in order to make a quadrilateral. If opposite angles total 180°, the four corners are concyclic. The other pair follows, because all four angles total 360°.
- 65°, 100°, 115°, 80°: 65 + 115 = 180. Concyclic.
- 70°, 100°, 105°, 85°: 70 + 105 = 175. Not concyclic.
Worked with points
Take (0, 0), (4, 0), (4, 3) and (0, 3). The first three make a right angle at (4, 0). So the side from (0, 0) to (4, 3) is a diameter. Its length is √(4² + 3²) = 5, so the radius is 2.5. The centre is its midpoint, (2, 1.5). Check (0, 3): the gaps are 2 and 1.5. So 2² + 1.5² = 6.25, and 2.5² is 6.25 too. The fourth point is also 2.5 from the centre. All four are concyclic.
Change the fourth point to (0, 4). The first three points are the same, so the circle is too. The gaps to (0, 4) are 2 and 2.5. So 2² + 2.5² = 10.25, and √10.25 is about 3.2. That is not the radius, 2.5. No circle passes through all four now.
| Corners | Fourth point's distance | Concyclic? |
|---|---|---|
| (0,0) (4,0) (4,3) (0,3) | 2.5, radius 2.5 | Yes |
| (0,0) (4,0) (4,3) (0,4) | about 3.2, radius 2.5 | No |
Each card gives four corners. Decide: one circle, or not?
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