Editing Similarity (geometry) (section)

==Similar triangles==
Two triangles, {{math|△''ABC''}} and {{math|△''A'B'C'''}} are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of [[corresponding sides]] are [[proportionality (mathematics)|proportional]].{{sfn|Sibley|1998|p=35}} It can be shown that two triangles having congruent angles (''equiangular triangles'') are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem.{{sfn|Stahl|2003|p=127. This is also proved in [[Euclid's Elements|Euclid's ''Elements'']], Book VI, Proposition 4}} Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle". Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.<ref>For instance, {{harvnb|Venema|2006|p=122}} and {{harvnb|Henderson|Taimiņa|2005|p=123}}.</ref>

There are several criteria each of which is necessary and sufficient for two triangles to be similar:

*Any two pairs of angles are congruent,<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book VI, Proposition 4.</ref> which in Euclidean geometry implies that all three angles are congruent:{{efn|This statement is not true in [[non-Euclidean geometry]] where the triangle angle sum is not 180 degrees.}}

::If {{math|∠''BAC''}} is equal in measure to {{math|∠''B'A'C',''}} and {{math|∠''ABC''}} is equal in measure to {{math|∠''A'B'C',''}} then this implies that {{math|∠''ACB''}} is equal in measure to {{math|∠''A'C'B'''}} and the triangles are similar.

*All the corresponding sides are proportional:<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book VI, Proposition 5.</ref>

<math display=block>\frac{\overline{AB}}{\overline{A'B'}} = \frac{\overline{BC}}{\overline{B'C'}} = \frac{\overline{AC}}{\overline{A'C'}}.</math> 
:This is equivalent to saying that one triangle (or its mirror image) is an [[homothetic transformation|enlargement]] of the other.

*Any two pairs of sides are proportional, and the angles included between these sides are congruent:<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book VI, Proposition 6.</ref>
<math display=block> \frac{\overline{AB}}{\overline{A'B'}} = \frac{\overline{BC}}{\overline{B'C'}}, \quad \angle ABC \cong \angle A'B'C'.</math>
:This is known as the SAS similarity criterion.{{sfn|Venema|2006|p=143}} The "SAS" is a mnemonic: each one of the two S's refers to a "side"; the A refers to an "angle" between the two sides.

Symbolically, we write the similarity and dissimilarity of two triangles {{math|△''ABC''}} and {{math|△''A'B'C'''}} as follows:<ref name=PL>{{cite book|last1=Posamentier|first1=Alfred S.|authorlink=Alfred S. Posamentier|last2=Lehmann|first2=Ingmar|year=2012|title=[[The Secrets of Triangles]]|publisher=Prometheus Books|page=22}}</ref>

<math display=block>\begin{align}
\triangle ABC &\sim \triangle A'B'C' \\
\triangle ABC &\nsim \triangle A'B'C' 
\end{align}</math>

There are several elementary results concerning similar triangles in Euclidean geometry:{{sfn|Jacobs|1974|pp=384–393}}
* Any two [[equilateral triangle]]s are similar.
* Two triangles, both similar to a third triangle, are similar to each other ([[transitive relation|transitivity]] of similarity of triangles).
* Corresponding [[altitude (triangle)|altitudes]] of similar triangles have the same ratio as the corresponding sides.
* Two [[right triangle]]s are similar if the [[hypotenuse]] and one other side have lengths in the same ratio.<ref>{{cite book|last=Hadamard |first=Jacques|author-link=Jacques Hadamard|year=2008|title=Lessons in Geometry, Vol. I: Plane Geometry|publisher=American Mathematical Society|isbn=978-0-8218-4367-3|at=Theorem 120, p. 125|url=https://books.google.com/books?id=SaZwAAAAQBAJ&pg=PA125}}</ref> There are several equivalent conditions in this case, such as the right triangles having an acute angle of the same measure, or having the lengths of the legs (sides) being in the same proportion.

Given a triangle {{math|△''ABC''}} and a line segment {{math|{{overline|''DE''}}}} one can, with a [[straightedge and compass construction|ruler and compass]], find a point {{mvar|F}} such that {{math|△''ABC'' ~ △''DEF''}}. The statement that point {{mvar|F}} satisfying this condition exists is [[John Wallis#Geometry|Wallis's postulate]]<ref>Named for [[John Wallis]] (1616–1703)</ref> and is logically equivalent to Euclid's [[parallel postulate]].{{sfn|Venema|2006|p=122}} In [[hyperbolic geometry]] (where Wallis's postulate is false) similar triangles are congruent.

In the axiomatic treatment of Euclidean geometry given by [[George David Birkhoff]] (see [[Birkhoff's axioms]]) the SAS similarity criterion given above was used to replace both Euclid's parallel postulate and the SAS axiom which enabled the dramatic shortening of [[Hilbert's axioms]].{{sfn|Venema|2006|p=143}}

Similar triangles provide the basis for many [[synthetic geometry|synthetic]] (without the use of coordinates) proofs in Euclidean geometry. Among the elementary results that can be proved this way are: the [[angle bisector theorem]], the [[geometric mean theorem]], [[Ceva's theorem]], [[Menelaus's theorem]] and the [[Pythagorean theorem]]. Similar triangles also provide the foundations for [[trigonometry|right triangle trigonometry]].{{sfn|Venema|2006|p=145}}