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...m]] and a square version of the [[Napoleon's theorem]]. All three of these theorems are just a special case of [[Petr–Douglas–Neumann theorem]]. ...e vertex of the square distant from both triangles and the vertices of the triangles distant from the square is equilateral.<ref>http://www.cut-the-knot.org/Cur ...

{{Short description|Geometry theorem relating to triangles}} ...o sides of another triangle, and the [[Congruence (geometry)#Congruence of triangles|included angle]] of the first is larger than the included angle of the seco ...

...mula]]'' can be thought as a special case of the Brahmagupta's formula for triangles. ...roaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula. ...

[[Category:Theorems about triangles and circles]] ...

...nt on [[inellipse]]s of triangles. From a projective point of view the two triangles <math>P_1P_3P_5</math> and <math>P_2P_4P_6</math> lie perspectively with ce [[Category:Theorems in projective geometry]] ...

Euler's inequality, in the form stating that, for all triangles inscribed in a given circle, the maximum of the radius of the inscribed cir *[[Poncelet's closure theorem]], showing that there is an infinity of triangles with the same two circles (and therefore the same ''R'', ''r'', and ''d'') ...

...ter how one triangulates a cyclic ''quadrilateral'', the sum of inradii of triangles is constant. [[Category:Theorems about triangles and circles]] ...

...oint ''F'' the triangles ''FPX'', ''FQY'', and ''FRZ'' are similar to the triangles ''AOI, BOI, COI'' respectively.<ref name=Kiss/>{{rp|Propos. 4}} Its [[barycentric coordinate system#Barycentric coordinates on triangles|barycentric coordinates]] are<ref name=Kiss/> ...

===Proof using similar triangles=== ...ngle ABD</math>. Because the ratios between corresponding sides of similar triangles are all equal, it follows that <math>|AB|/|AC_2| = |BD|/|CD|</math>. Howeve ...

[[File:Congruent_non-congruent_triangles.svg|thumb|upright=1.2|The two triangles on the left are congruent. The third is [[Similarity (geometry)|similar]] t *Third, ''rotate'' the translated figure about the matched vertex until one pair of [[corresponding sides]] matches. ...

{{short description|Two triangles are in perspective axially if and only if they are in perspective centrally ...f perspectivity. The lines which run through corresponding vertices on the triangles meet at a point called the center of perspectivity. Desargues's theorem sta ...

...orem in 1746.<ref>{{citation|first=Matthew|last=Stewart|title=Some General Theorems of Considerable Use in the Higher Parts of Mathematics|year=1746|publisher= ...|1=cos ''θ' ''= −cos ''θ''}}. Applying the law of cosines in the two small triangles using angles {{mvar|θ}} and {{mvar|θ'}} produces ...

...gon into triangles and [[trapezoids]], which in turn can be converted into triangles. Secondly, each of these triangles can be transformed into a right triangle and subsequently into a [[rectangl ...

...ric Delights'', [[Princeton University Press]], 2002.</ref> is a statement about the relationship between the tangents of two angles of a [[triangle]] and t ...yani|Ibn Muʿādh al-Jayyānī]] also described the law of tangents for planar triangles in the 11th century.<ref name=Debarnot /> ...

For acute triangles, the feet of the altitudes all fall on the triangle's sides (not extended). ==Theorems== ...

...st=Andrew P.|last=Guinand|title=Euler lines, tritangent centers, and their triangles|journal=[[American Mathematical Monthly]]|volume=91|issue=5|jstor=2322671|y ...oncurrent lines|concurrent]] at the nine-point center common to all of the triangles.<ref name="ac">Altshiller-Court, Nathan, ''College Geometry'', Dover Public ...

...elaus's theorem''', named for [[Menelaus of Alexandria]], is a proposition about [[triangle]]s in [[plane geometry]]. Suppose we have a triangle {{math|△''A ...t|signed lengths of segments]], which provides some additional information about the relative order of collinear points. Here, the length {{mvar|{{overline| ...

==Theorems on books== ...

...riosity with regards to [[geometry]] and [[trigonometry]] – particularly [[triangles]] and [[rectangles]]. These were the shapes which provided the most questio Proof as we know it came about with one specific question: “what is a proof?” Traditionally, a proof is a ...

...if all the trisectors are intersected, one obtains four other equilateral triangles. The law of sines applied to triangles <math>AYC</math> and <math>AZB</math> yields ...