What is an arbitrary point?

Question

Firstly, how is an arbitrary point defined in plane geometry?

I came across many proofs which use an arbitrary point to prove something which is true for all points.

For eg: Prove that the tangent to a circle is perpendicular to the line passing through the point of contact and the circle's centre.

the line passing through centre and point of contact of a tangent is perpendicular to the circle's centre.

Is there any proof of this technique? If yes, then how effective is this technique when it comes to proofs?

Answer 1

If you truly want to understand this 'proof mechanism', you need to learn a deductive system for FOL (first-order logic), such as this one. In that system, what you are asking about is then simply as follows.

When you want to prove a statement of the form "$∀x{∈}S \ ( \ Q(x) \ )$, you create a subcontext (by the $∀$sub rule), writing "Given $x{∈}S$:" and starting to reason under that header. This means that you are given an unknown member $x$ of type $S$, and you wish to prove $Q(x)$. If your reasoning works (despite you not knowing anything about $x$ except $x{∈}S$), then you would have shown that just outside that subcontext you can assert "$Q(x)$ for every $x{∈}S$.". This final deductive step is indeed permitted by the $∀$intro rule.

In your specific example, the theorem you want to prove is actually:

  For every circle $C$ with centre $O$, and every line $L$ tangent to $C$ at $P$,
    we have that $L$ is perpendicular to $OP$.

This is clearly a $∀$-statement, and so you actually need the $∀$intro rule in order to prove it! That is, it is not merely an "effective" technique, but a necessary step. What I said applies in some equivalent form to any other deductive system for FOL, and not just the linked system. And FOL is the basis for almost all of modern mathematics. So you do need to understand it if you want to truly understand mathematics.

Answer 2

As C-RAM's comment suggested, "arbitrary" means undetermined; not assigned to a specific value.

It is better not to think of it in terms of being arbitrary or determined, but rather just simply as a domain of values for which there is no loss of generality to the accuracy of a statement.

For example, instead of saying for any arbitrary point of tangency on a circle, just say for all points (a,b) on the circle there exists a point of tangency to which the angle made with the circles center is a right angle.

By doing this, we can avoid any dispute about what it "really" means to be arbitrary, although that can be a further topic of debate when it is used in other contexts such as "an arbitrarily small length".