Have you ever stopped to ask yourself where geometry comes from? How did we create or discover so much structure? Evidence of ancient uses of geometry lies all around the world. From the Pyramids$^2$ constructed in Egypt to the Great Wall of China, it is clear that geometry was used and understood as early as 2500 B.C. Furthermore, even the creation of the wheel denotes a basic understanding of geometric principles. However, up until about 300 B.C.$^4$ geometric principles were not recorded using consistent practices.
Around 300 B.C. a Greek mathematician by the name of Euclid saw this issue and set out to record geometry as it was understood. More than just recording geometry, Euclid began with 5 basic axioms and used these axioms to prove each element of geometry. It is a commonly known fact that the first book printed on the printing press was the Bible, less commonly known is that the second book printed on the printing press was Euclid's record of geometry. A young Isaac Newton$^3$ was often found reading Euclid's book. Even Abraham Lincoln would "read Euclid by the light of a candle after others had dropped off to sleep"$^3$ as a young lawyer. The name of this book was "Elements" and, with the exception of the Bible, it is not hyperbole to claim that "Elements" could be the most influential book in human history.
Before understanding theorems and their significance and applications of triangles, we need to briefly discuss some key elements of geometric structure. Euclid based "Elements" on 5 axioms that he claimed were necessary to prove all other aspects of Geometry. The existence of these 5 axioms is what defines Euclidean Geometry. You may have heard of Non-Euclidean Geometry. Non-Euclidean Geometry is geometry in which Euclid's 5th axiom is not assumed to be true.
Euclid expressed everything in geometry as a set of points. Using this idea he developed spaces, planes, lines, rays, line segments, and angles. From these, he continued to develop triangles and other shapes such as squares and circles. More than just developing these elements of geometry, Euclid proved theorems connecting conditions of these elements to specific results, findings and properties. This website specifically considers the results that Euclid defined for triangles.
2: Wier, S. K. (2008, December 22). Insight from Geometry and Physics into the Construction of Egyptian Old Kingdom Pyramids | Cambridge Archaeological Journal. In Cambridge Core. Retrieved November 1, 2022, from https://www.cambridge.org/core/journals/cambridge-archaeological-journal/article/abs/insight-from-geometry-and-physics-into-the-construction-of-egyptian-old-kingdom-pyramids/DA7ED1D211A218D73DFCD0E08463D237
3: Carl Lee. (1997, September 11). Euclid’s Elements and Abraham Lincoln. https://www.ms.uky.edu%7Elee/ma341fall97/quotes/node2.html
4: Artmann, B. (2020, September 10). Euclidean geometry | Definition, Axioms, & Postulates. Encyclopedia Britannica.https://www.britannica.com/science/Euclidean-geometry