Equal angles on one side put four points on a circle
Show on a big screenA wall runs from A to B. At C, angle ACB is 53°. At D, angle ADB is 53° too. Do all four points share one circle?
- From C, the segment AB makes 53°.
- The circle through A, B and C.
- D on the circle sees AB at 53° too.
- D inside the circle sees AB at 77°. Bigger.
- D outside the circle sees AB at 44°. Smaller.
The rule
Yes, if C and D lie on one side of line AB.
Rule: AB makes equal angles at C and D. C and D are on the same side of line AB. Then A, B, C and D lie on one circle.
Four points on one circle are concyclic.
Why? A, B and C are not in a line. So one circle passes through them. Test D against that circle. Suppose D is inside it, on the same side of line AB. Then angle ADB is bigger than angle ACB. Suppose D is outside it. Then angle ADB is smaller. But D makes the same angle as C. So D is on the circle.
Let A = (0, 0), B = (8, 0) and C = (−1, 3). Take the circle through A, B and C. Its centre is (4, 3) and its radius is 5. Check: the centre is 5 from A, B and C. Angle ACB is 53.1°.
| D | Where it sits | Angle ADB |
|---|---|---|
| (7, 7) | On the circle | 53.1° |
| (4, 5) | Inside the circle | 77.3° |
| (7, 9) | Outside the circle | 44.2° |
| (4, 8) | On the circle | 53.1° |
Watch out!
Both points must be on the same side of line AB. On opposite sides, equal angles fit one circle only at 90°.
Slide D up the line. Watch the angle at D.
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