Constructing the inverse of a number geometrically.

Question

this picture:

shows a way to construct the inverse of a number $a\ge1$. but how can we construct for a number that is less than 1?

My try::

1. Q1: is my try correct?
2. Q2: how to prove them both?

Answer 1

Given the diagram as labeled ...

... we consider that construction occurred as follows:

Starting with point $P$ (or point $Q$) on $\overrightarrow{OR}$, construct $\overleftrightarrow{AP}$ (or $\overleftrightarrow{BQ}$) and let $C$ be the point where this line meets the unit circle. Then $\overleftrightarrow{BC}$ (or $\overleftrightarrow{AC}$) determines the point $Q$ (or $P$) on $\overrightarrow{OR}$.

Now, because $\angle ACB$ is inscribed in a semi-circle, it is a right angle by Thales' Theorem. Consequently, $\angle P \cong \angle B$ (as each is the complement of $\angle A$), so that $\triangle POA \sim \triangle BOQ$ and we can write $$\frac{|\overline{OP}|}{|\overline{OA}|} = \frac{|\overline{OB}|}{|\overline{OQ}|} \qquad\to\qquad \frac{|\overline{OP}|}{1} = \frac{1}{|\overline{OQ}|}$$

This proves the reciprocal relation. $\square$

Note: Even when the circle doesn't have unit radius, the relationship involves the geometric mean $$|\overline{OR}|^2 \;=\; |\overline{OP}|\;|\overline{OQ}|$$ which is important for the study of inversive geometry and such. The construction given is a nice companion to the more-common (to me) one involving the chord between points of tangency.