How to construct the product and the inverse of a given number (using ruler and compass)?

Question

I am trying to prove that the set of all ruler-and-compass constructible points is a subfield of $R$. I have one difficulty though:

• Given two numbers $\sqrt{a}$ and $\sqrt{b}$ , how can I construct their product and their inverse with rule and compass?

Answer 1

Think similar triangles. Make a $1, \sqrt a$ right triangle and then a similar one that is $\sqrt b, \sqrt{ab}$

Answer 2

One way to construct a product, an inverse and also a square root is to use Intersecting chords theorem:

But we also need to know a segment of unit length.

For example, if $|AB|=a$, $|BC|=b$, and a unit length $|BD|=1$, then $a\,b=|BE|$. But if a unit length is instead $|BE|=1$, then $a\,b=|BD|$.

The same construction can be used to get an inverse of a number as well as a square root.

Answer 3

In the next picture it is $AM×BM=1$ Proof: The figure formed by the tangent to the unitary circle and the two parallel lines tangent to the unitary circle itself, can be completed to a rhombus whose sides touch the unitary circle, we know that the center of the circle tangent to the sides of a rhombus is the point of intersection of its diagonals and we know that the two diagonals of the rhombus are perpendicular

So $AO⊥BO$.

We know that the tangent to a circle at a point is perpendicular to the radius of the circle that includes the point of tangency

So $OM⊥AB$.

$OM$ is the height of the hypotenuse in a right triangle $∆AOB$, so: $OM^2=AM⋅BM$ $⇒AM×BM=1$