Propositions
From true
Propositions (or theorems) are statements which are provable by using the definitions, postulates, and common notions as well as any previous propositions which have been proven.
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[edit] Proposition 1: You can create an equilateral triangle on any given line segment.
This proposition uses two circles, each drawn with the same radius, to find a point equidistant from the ends of the given line segment. Connecting that point to the ends of the line segment creates an equilateral triangle.
The proof of this proposition is based only on the following:
- describing circles (Postulate 3: One can describe a circle with any center and radius.)
- drawing lines (Postulate 1: One can draw a straight line from any point to any point.)
- the definition of a circle (Definitions 15 and 16: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another. And the point is called the center of the circle.)
- and common notion 1 (Things which equal the same thing also equal one another.)
[edit] Other propositions
We also went over three propositions which rely on the congruence theorems (side-side-side, side-angle-side, and angle-side-angle). All three end by using the idea that corresponding parts of congruent triangles are equal.
[edit] Proposition A (or Proposition 9): You can bisect any given angle.
To prove this, start by constructing a circle on the vertex of the given angle. That will give you two points on which to construct an equilateral triangle (Proposition 1).
[edit] Proposition B (or Proposition 10): You can bisect any given line segment.
To prove this, you again start with an equilateral triangle (Proposition 1) and then you can bisect the angle at the newly formed vertex.
[edit] Proposition C (see Propositions 11 and 12): You can draw a line perpendicular to any given line.
To prove this, you just need to add a few more statements to the same proof used for Proposition B.
