In the master thesis we introduce and present the standard geometric constructions with ruler and compass. We set the analytical criteria for constructibility and show how to construct new geometric objects from already given ones. Then we show that all ruler and compass numbers form an Euclidean field.
In the next chapter we switch to the non-standard geometric constructions with ruler only, and we pick a starter set with four arbitrary points. We use a theorem from elementary geometry - the Trapezoid Theorem, that gives us a construction to bisect the segments on the parallel lines, and its Corollary, that gives us a construction of a parallel line that intersects the segment bisecting line. We continue with definitions of a ruler point, a ruler line, a ruler circle and a ruler number. We show which constructions are possible with ruler alone and characterize the field of ruler numbers.
In the third chapter we introduce the geometric constructions with ruler and dividers. We also present other geometric tools that are similar to dividers, such as rusty dividers, angle bisector and the cannon.
We show that all ruler and dividers numbers form a Pythagorean field.
Next, we move on to the main chapter of our thesis, the ruler and circle constructions. We pick a different starter set and we define a ruler and circle point and a ruler and circle line. We proclaim the main theorem - The Poncelet-Steiner Theorem and we demonstrate the analytic methods to prove the theorem.
In the last chapter we present the basic Steiner constructions. With a proof of all these constructions, we also prove the Poncelet-Steiner Theorem in a different, synthetic way.
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