Editing Euclidean geometry (section)

== Some important or well known results ==
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File:pons_asinorum_dzmanto.png|The ''[[pons asinorum]]'' or ''bridge of asses theorem'' states that in an isosceles triangle, α&nbsp;=&nbsp;β and γ&nbsp;=&nbsp;δ.
File:Sum_of_angles_of_triangle_dzmanto.png|The ''triangle angle sum theorem'' states that the sum of the three angles of any triangle, in this case angles α, β, and γ, will always equal 180 degrees.
File:Pythagorean.svg|The ''[[Pythagorean theorem]]'' states that the sum of the areas of the two squares on the legs (''a'' and ''b'') of a right triangle equals the area of the square on the hypotenuse (''c'').
File:Thales' Theorem Simple.svg|''[[Thales' theorem]]'' states that if AC is a diameter, then the angle at B is a right angle.
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===Pons asinorum===
The [[pons asinorum]] (''bridge of asses'') states that ''in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another''.<ref>Euclid, book I, proposition 5, tr. Heath, p.&nbsp;251.</ref> Its name may be attributed to its frequent role as the first real test in the ''Elements'' of the intelligence of the reader and as a bridge to the harder propositions that followed. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross.<ref>Ignoring the alleged difficulty of Book I, Proposition 5, [[T. L. Heath|Sir Thomas L. Heath]] mentions another interpretation. This rests on the resemblance of the figure's lower straight lines to a steeply inclined bridge that could be crossed by an ass but not by a horse: "But there is another view (as I have learnt lately) which is more complimentary to the ass. It is that, the figure of the proposition being like that of a trestle bridge, with a ramp at each end which is more practicable the flatter the figure is drawn, the bridge is such that, while a horse could not surmount the ramp, an ass could; in other words, the term is meant to refer to the sure-footedness of the ass rather than to any want of intelligence on his part." (in "Excursis II", volume 1 of Heath's translation of ''The Thirteen Books of the Elements'').</ref>

===Congruence of triangles===
[[File:Congruent triangles.svg|thumb|right|Congruence of triangles is determined by specifying two sides and the angle between them (SAS), two angles and the side between them (ASA) or two angles and a corresponding adjacent side (AAS). Specifying two sides and an adjacent angle (SSA), however, can yield two distinct possible triangles unless the angle specified is a right angle.]]

Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26). Triangles with three equal angles (AAA) are similar, but not necessarily congruent. Also, triangles with two equal sides and an adjacent angle are not necessarily equal or congruent.

===Triangle angle sum===
The sum of the angles of a triangle is equal to a straight angle (180 degrees).<ref>Euclid, book I, proposition 32.</ref> This causes an equilateral triangle to have three interior angles of 60 degrees. Also, it causes every triangle to have at least two acute angles and up to one [[obtuse angle|obtuse]] or [[right angle]].

===Pythagorean theorem===
The celebrated [[Pythagorean theorem]] (book I, proposition 47) states that in any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

===Thales' theorem===
[[File:Congruentie.svg|thumb|An example of congruence. The two figures on the left are congruent, while the third is [[Similarity (geometry)|similar]] to them. The last figure is neither. Congruences alter some properties, such as location and orientation, but leave others unchanged, like [[distance]] and [[angle]]s. The latter sort of properties are called [[invariant (mathematics)|invariant]]s and studying them is the essence of geometry.|218x218px]]
[[Thales' theorem]], named after [[Thales of Miletus]] states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Cantor supposed that Thales proved his theorem by means of Euclid Book I, Prop. 32 after the manner of Euclid Book III, Prop. 31.<ref>Heath, p.&nbsp;135. [https://books.google.com/books?id=drnY3Vjix3kC&pg=PA135 Extract of page 135].</ref><ref>Heath, p.&nbsp;318.</ref>

===Scaling of area and volume===
In modern terminology, the area of a plane figure is proportional to the square of any of its linear dimensions, <math>A \propto L^2</math>, and the volume of a solid to the cube, <math>V \propto L^3</math>. Euclid proved these results in various special cases such as the area of a circle<ref>Euclid, book XII, proposition 2.</ref> and the volume of a parallelepipedal solid.<ref>Euclid, book XI, proposition 33.</ref> Euclid determined some, but not all, of the relevant constants of proportionality. For instance, it was his successor [[Archimedes]] who proved that a sphere has 2/3 the volume of the circumscribing cylinder.<ref>Ball, p.&nbsp;66.</ref>