Why geometry refuses to define its first words
Open any dictionary and look up a word. The definition is made of other words, which you can also look up, and those of still others. Either the chain loops back on itself or it runs forever. Mathematics, which wants every claim to rest on something solid, cannot live with that. So it makes a bold, honest move: it picks a tiny handful of words and agrees never to define them at all.
These are the undefined terms. In plane geometry there are essentially three of them — point, line, and plane — together with a relation like 'lies on' that ties them together. We do not say what a point is. We do not draw a tiny picture and claim that is the point. Instead we simply use the word, and we pin down its behaviour later through rules. An undefined term is not a gap in the theory; it is the bedrock the theory is honest enough to stand on.
A point: a place with no size at all
Picture the sharpest dot a pencil can make. Now imagine sharpening it forever — narrower, finer, until it has no width and no height left. What remains is a pure location, an answer to the question 'where?' with nothing left over to ask 'how big?'. That is a point: it has position but zero size. We name points with capital letters: A, B, C. On paper the dot we draw is only a stand-in, a friendly lie, because a true point is too small to see.
Because a point has no size, it has no parts and nothing inside it. This is exactly Euclid's very first line in the Elements: 'A point is that which has no part.' He is not really defining it — you cannot build 'point' out of simpler words — he is describing the picture he wants in your head. Two distinct points are just two different places, and a single point all by itself can never be 'between', 'left of', or 'longer than' anything. It only becomes interesting once a second point joins it.
A line: straight, endless, and only one through two points
Mark two points, A and B. Pull a thread taut between them and keep pulling past both ends, never letting it sag or curve, forever. That perfectly straight, endless track is a line. In geometry the bare word 'line' always means a straight line, and it has no thickness and no end — the arrowheads we draw on it are just a reminder that it keeps going. We write it 'line AB', naming it by any two of its points.
Here is the rule that makes a line a line, and it is the first real piece of geometry you can hold onto: through any two distinct points there passes exactly one line. 'At least one' says you can always connect two places; 'exactly one' says there is never a choice — A and B pin their line down completely. This single sentence is one of Euclid's postulates, and it is doing heavy lifting already. From it follows the fact that two different lines can share at most one point: if they shared two, those two points would force the lines to be the very same line.
Three or more points that all sit on one single line are called collinear. Any two points are automatically collinear — there is always a line through them — so the idea only earns its keep at three points, where it becomes a genuine question: do A, B, and C all lie on one common line, or do they not? That little question is the seed of nearly every theorem to come.
A plane: the flat tabletop that never ends
Now lift your gaze off the line. Imagine a perfectly flat tabletop — but with no edges, no thickness, stretching out forever in every direction. That endless flatness is a plane. A sheet of paper, a calm pond, a wall: each is a finite scrap of a plane, and again the real thing is too big and too thin to see. Most of beginning geometry lives entirely inside one such plane, which is why it is called plane geometry.
Just as two points pin down a line, three points pin down a plane — provided the three are not collinear. Through any three non-collinear points there passes exactly one plane. This is why a three-legged stool never wobbles while a four-legged chair on an uneven floor can: three feet always settle onto one definite plane, but a fourth point may insist on floating above or below it. Points that do share a common plane are called coplanar, the planar cousin of collinear.
Putting the three together
Point, line, and plane are nested in size — a point has zero dimensions, a line one, a plane two — and they hook together through a few quiet rules of incidence: rules about what lies on what. Two points determine a line; three non-collinear points determine a plane; if two points of a line lie in a plane, then the whole line lies in that plane. None of these are proved. They are the agreed starting moves, and from them everything else will be argued.
A . . . . . . . B <-- two points
<---A-------B---> <-- line AB: the one line through them
C
\
\ A, B, C non-collinear
A---+---B --> one plane holds all three
So you now hold geometry's three starting words, left undefined on purpose, plus the first few rules that govern them. In the next guide we make those rules official: we separate the things we accept without proof from the things we earn by proof, and meet the cast of postulates, definitions, and theorems that run the rest of the game.