# Constructible Points and Numbers

- Author:
- Ku, Yin Bon (Albert)

In order to analyse the construction power of straightedge and compass, we need to use coordinate geometry.

**Constructible points**are simply points that can be produced by Euclidean constructions. A more rigorous definition of Euclidean construction is needed:

- We should get rid of any randomness - we are not allowed to select an "arbitrary point" in the process of construction. Any new point should come from an intersection between lines or circles that are drawn previously.
- Obviously, we need to have at least two given points to start a construction. By convention, and and the two initially given points.

**constructible number**if is a constructible point. We have the following useful result:

is a constructible point if and only if and are constructible numbers.

**Exercise**In the following applet, construct the point from the points and using straightedge and Euclidean compass and hence show that it is a constructible point.

**Note**: No "Well done!" message will appear for this exercise.