Three points in one line, or very nearly
Three pegs look lined up from one end. From the side, the middle one is not on the line.
- Start at (1, 2).
- Hop to (3, 6): 2 across, 4 up.
- Hop again to (6, 12): 3 across, 6 up.
- Same direction both times: one straight line.
- Start at (1, 2).
- Hop to (3, 6): 2 across, 4 up.
- Hop again to (6, 11): 3 across, 5 up.
- Different direction: not one straight line.
Try it first. Move C up or down. Make the two products equal.
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The rule
Take three points in order. A hop is the move from one point to the next. Points lie on one line when the hops are in proportion. That means up ÷ across is the same for both hops. A hop straight up has 0 across, and dividing by 0 fails. So do not divide. Cross-multiply: first across × second up, and first up × second across. Equal products mean one straight line.
Take (1, 2), (3, 6) and (6, 12). First hop: 2 across, 4 up. Second hop: 3 across, 6 up. Cross-multiply: 2 × 6 = 12, and 4 × 3 = 12. Equal, so the three points lie on one line.
Now move the last point to (6, 11). The second hop becomes 3 across, 5 up. Cross-multiply: 2 × 5 = 10, against 4 × 3 = 12. 10 is not 12, so these three points miss the line.
| The three points | Hop test | In a line? |
|---|---|---|
| (1, 2), (3, 6), (6, 12) | 2 × 6 = 12, 4 × 3 = 12 | yes |
| (0, 0), (2, 3), (6, 9) | 2 × 6 = 12, 3 × 4 = 12 | yes |
| (1, 2), (3, 6), (6, 11) | 2 × 5 = 10, 4 × 3 = 12 | no |
| (0, 0), (5, 3), (10, 7) | 5 × 4 = 20, 3 × 5 = 15 | no |
A second test uses distances. Take A = (1, 2), B = (3, 6) and C = (6, 11). If B is between A and C, then AB + BC = AC. Here AB + BC is about 10.303. But AC is about 10.296. A gap of 0.007 is still a miss.