Proof: By Euclid
(related to Corollary: Existence of Parallel Straight Lines)
- Extend the segments \(\overline{BA}\) and \(\overline{CF}\) to straight lines \(BA\) and \(CF\).
- Assume that \(BA\) and \(CF\) meet somewhere at a point \(X\). Then, the triangle \(\triangle{CAX}\) would have an exterior angle \(\angle{BAC}\) equal to the interior angle \(\angle{XCA}\), in contradiction to Prop 1.16, according to which it must be greater than the interior angle.
- Therefore, the hypothesis \(BA\) and \(CF\) would meet at a point \(X\) must be false. In other words, the straight lines do not have any point in common and exist.
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References
Adapted from CC BY-SA 3.0 Sources:
- Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014
Adapted from (Public Domain)
- Casey, John: "The First Six Books of the Elements of Euclid"
