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?
Best Answer
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".