Constructing tangents to a circle from an external point.

Question

I learned a method of constructing tangents to a circle O from an external point P which involves (a) drawing segment OP, (b) finding the midpoint M of OP, (c) constructing circle M with radius MP, (d) drawing a tangent from P to the intersections of circles O and M.

 

My question is simple.  Why do we need to go through all these steps?  Why can't we just use a straightedge to connect P to circle O such that each line crosses circle O at only one point?  It seems like a lot of extra steps to achieve the same result.

Answer 1

A point does not have length or width (definition you should have learned).

Those "points" created by your drawing instrument do have length and width.

Although you may not be able to see them, your drawn line may intersect two (or more) of them.