Lines through points

Rihan marks one point on a piece of paper. How many lines can he draw through it? As many as he likes! Lines can pass through one point in every direction, like the spokes of a wheel through its centre, with no limit.

Sheetal marks two points. How many different lines pass through both? Only one. Any two points fix a unique line. Try to draw a second line through both points, and it falls right on top of the first.

Now mark three points A, B and C that are not on one line, and draw every line through a pair of them. Each pair gives one line: AB, BC and CA. That is 3 lines.

With four points A, B, C, D, where no three are on one line, count the pairs carefully so you do not count any twice:

• from A: AB, AC, AD (3 new lines);
• from B: BC, BD (BA is the same line as AB, so 2 new);
• from C: CD (1 new).

Altogether 3 + 2 + 1 = 6 lines.