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''trigonometry in the Euclidean plane.''
 
''trigonometry in the Euclidean plane.''
  
The elements of a triangle, its sides $  a, b, c $
+
The elements of a triangle, its sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p0728101.png" /> and its angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p0728102.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p0728103.png" /> opposite to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p0728104.png" />, etc.), are related in various ways. In the Euclidean plane the most important relations are the angle sum formula
and its angles $  A, B, C $(
 
$  A $
 
opposite to $  a $,  
 
etc.), are related in various ways. In the Euclidean plane the most important relations are the angle sum formula
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p0728105.png" /></td> </tr></table>
A + B + C  = \pi
 
$$
 
  
 
(angles in radians), and the triangle inequalities
 
(angles in radians), and the triangle inequalities
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p0728106.png" /></td> </tr></table>
a + b  > c ,\ \
 
b + c  > a ,\ \
 
c + a  > b .
 
$$
 
  
These inequalities are necessary and sufficient for three segments of positive length $  a, b, c $
+
These inequalities are necessary and sufficient for three segments of positive length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p0728107.png" /> to form the sides of a triangle.
to form the sides of a triangle.
 
  
 
Another relation is the [[Cosine theorem|cosine theorem]]:
 
Another relation is the [[Cosine theorem|cosine theorem]]:
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p0728108.png" /></td> </tr></table>
c  ^ {2}  = a  ^ {2} + b  ^ {2} - 2ab  \cos  C.
 
$$
 
  
In particular, when $  C = \pi /2 $,  
+
In particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p0728109.png" />, the triangle is right-angled, and the cosine theorem becomes Pythagoras' theorem (cf. [[Pythagoras theorem|Pythagoras theorem]])
the triangle is right-angled, and the cosine theorem becomes Pythagoras' theorem (cf. [[Pythagoras theorem|Pythagoras theorem]])
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281010.png" /></td> </tr></table>
c  ^ {2}  = a  ^ {2} + b  ^ {2} .
 
$$
 
  
 
In such a right-angled triangle,
 
In such a right-angled triangle,
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281011.png" /></td> </tr></table>
  
 
In a general triangle, further relations are provided by the [[Sine theorem|sine theorem]]:
 
In a general triangle, further relations are provided by the [[Sine theorem|sine theorem]]:
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281012.png" /></td> </tr></table>
 
 
\frac{a}{\sin  A }
 
  =
 
\frac{b}{\sin  B }
 
  = \
 
  
\frac{c}{\sin  C }
+
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281013.png" /> is the radius of the circumcircle of the triangle (cf. [[Inscribed and circumscribed figures|Inscribed and circumscribed figures]]). A corollary of the sine theorem is the tangent formula
  = 2R ,
 
$$
 
  
where  $  R $
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281014.png" /></td> </tr></table>
is the radius of the circumcircle of the triangle (cf. [[Inscribed and circumscribed figures|Inscribed and circumscribed figures]]). A corollary of the sine theorem is the tangent formula
 
  
$$
+
With the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281015.png" /> for the semi-perimeter of the triangle, the following half-angle formulas follow from the cosine theorem:
a-  
 
\frac{b}{a+}
 
b  = \
 
  
\frac{ \mathop{\rm tan} [( A- B)/2] }{ \mathop{\rm tan} [( A+ B)/2] }
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281016.png" /></td> </tr></table>
  = \
 
\mathop{\rm tan}  A-
 
\frac{B}{2}
 
  \mathop{\rm cotan} 
 
\frac{C}{2}
 
.
 
$$
 
  
With the notation  $  s = ( a+ b+ c)/2 $
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281017.png" /></td> </tr></table>
for the semi-perimeter of the triangle, the following half-angle formulas follow from the cosine theorem:
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281018.png" /></td> </tr></table>
\cos  ^ {2} 
 
\frac{A}{2}
 
  = s( s-  
 
\frac{a)}{bc}
 
,
 
$$
 
 
 
$$
 
\sin  ^ {2} 
 
\frac{A}{2}
 
  = ( s- b)( s-
 
\frac{c)}{bc}
 
,
 
$$
 
 
 
$$
 
\mathop{\rm tan}  ^ {2} 
 
\frac{A}{2}
 
  = ( s- b)( s-
 
\frac{c)}{s(}
 
s- a) .
 
$$
 
  
 
==Geometry of the triangle.==
 
==Geometry of the triangle.==
Among the many remarkable lines, points and circles connected with any triangle are the circumcircle with centre 0 and radius $  R $,  
+
Among the many remarkable lines, points and circles connected with any triangle are the circumcircle with centre 0 and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281019.png" />, the incircle and the three excircles with centres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281023.png" /> and radii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281027.png" /> (cf. [[Inscribed and circumscribed figures|Inscribed and circumscribed figures]]), the medians (cf. [[Median (of a triangle)|Median (of a triangle)]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281030.png" />, with the centroid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281031.png" /> as their common point, the inner bisectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281034.png" />, and the outer bisectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281037.png" />, the altitude lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281040.png" /> with the [[Orthocentre|orthocentre]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281041.png" /> as their common point, the Euler line (cf. [[Euler straight line|Euler straight line]]) through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281044.png" />, and the [[Nine-point circle|nine-point circle]] through the midpoints of the sides, the feet of the altitude lines, and the midpoints of the segments connecting the vertices of the triangle to its orthocentre. The nine-point circle has radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281045.png" />, its centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281046.png" /> is on the Euler line between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281048.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281049.png" />, and the nine-point circle touches the incircle and the three excircles (Feuerbach's theorem).
the incircle and the three excircles with centres $  I $,  
 
$  I _ {a} $,  
 
$  I _ {b} $,  
 
$  I _ {c} $
 
and radii $  r $,  
 
$  r _ {a} $,  
 
$  r _ {b} $,  
 
$  r _ {c} $(
 
cf. [[Inscribed and circumscribed figures|Inscribed and circumscribed figures]]), the medians (cf. [[Median (of a triangle)|Median (of a triangle)]]) $  m _ {a} $,  
 
$  m _ {b} $,  
 
$  m _ {c} $,  
 
with the centroid $  G $
 
as their common point, the inner bisectors $  AI $,  
 
$  BI $,  
 
$  CI $,  
 
and the outer bisectors $  I _ {b} I _ {c} $,  
 
$  I _ {c} I _ {a} $,  
 
$  I _ {a} I _ {b} $,  
 
the altitude lines $  h _ {a} $,  
 
$  h _ {b} $,  
 
$  h _ {c} $
 
with the [[Orthocentre|orthocentre]] $  H $
 
as their common point, the Euler line (cf. [[Euler straight line|Euler straight line]]) through $  O $,  
 
$  G $
 
and $  H $,  
 
and the [[Nine-point circle|nine-point circle]] through the midpoints of the sides, the feet of the altitude lines, and the midpoints of the segments connecting the vertices of the triangle to its orthocentre. The nine-point circle has radius $  R/2 $,  
 
its centre $  N $
 
is on the Euler line between $  G $
 
and $  H $
 
such that $  HN:  NG :  GO = 3: 1:  2 $,  
 
and the nine-point circle touches the incircle and the three excircles (Feuerbach's theorem).
 
  
With the notation $  ( ABC) $
+
With the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281050.png" /> for the area of the triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281051.png" />, the following relations are valid:
for the area of the triangle $  ABC $,  
 
the following relations are valid:
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281052.png" /></td> </tr></table>
( ABC)  =
 
\frac{1}{2}
 
ah _ {a}  =
 
\frac{1}{2}
 
bc  \sin  A  = \
 
  
\frac{abc}{4R\ }
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281053.png" /></td> </tr></table>
=
 
$$
 
 
 
$$
 
= \
 
r \cdot s  = r _ {a} ( s- a)  = \sqrt {s( s- a)( s- b)( s- c) } .
 
$$
 
  
 
It follows that, among others,
 
It follows that, among others,
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281054.png" /></td> </tr></table>
4R  = r _ {a} + r _ {b} + r _ {c} - r \  \textrm{ and } \ \
 
 
 
\frac{1}{r}
 
  =
 
\frac{1}{r _ {a} }
 
+
 
\frac{1}{r _ {b} }
 
+
 
\frac{1}{r _ {c} }
 
.
 
$$
 
  
 
Very remarkable is Morley's theorem: The points of intersection of the adjacent trisectors of the angles of any triangle form the vertices of an equilateral triangle. Indeed, a direct calculation shows that the sides of Morley's triangle have length
 
Very remarkable is Morley's theorem: The points of intersection of the adjacent trisectors of the angles of any triangle form the vertices of an equilateral triangle. Indeed, a direct calculation shows that the sides of Morley's triangle have length
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281055.png" /></td> </tr></table>
8R  \sin 
 
\frac{A}{3}
 
  \sin 
 
\frac{B}{3}
 
  \sin 
 
\frac{C}{3}
 
,
 
$$
 
  
which is symmetric in $  A $,  
+
which is symmetric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281058.png" />.
$  B $
 
and $  C $.
 
  
 
==The theorems of Ceva and Menelaus.==
 
==The theorems of Ceva and Menelaus.==
Let $  X, Y, Z $
+
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281059.png" /> be points on the (possibly extended) sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281060.png" /> of a triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281061.png" />. Then, by the [[Ceva theorem|Ceva theorem]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281064.png" /> are congruent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281065.png" /> (signed distances) and by the [[Menelaus theorem|Menelaus theorem]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281068.png" /> are collinear if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281069.png" />.
be points on the (possibly extended) sides $  a, b, c $
 
of a triangle $  ABC $.  
 
Then, by the [[Ceva theorem|Ceva theorem]], $  AX $,  
 
$  BY $
 
and $  CZ $
 
are congruent if and only if $  ( BX: XC)( CY:  YA)( AZ:  ZB)= 1 $(
 
signed distances) and by the [[Menelaus theorem|Menelaus theorem]], $  X $,  
 
$  Y $
 
and $  Z $
 
are collinear if and only if $  ( BX: XC)( CY:  YA)( AZ:  ZB) = - 1 $.
 
  
 
==Convex quadrangles.==
 
==Convex quadrangles.==
Ptolemy's theorem (cf. [[Ptolemeus theorem|Ptolemeus theorem]]): For any point $  P $
+
Ptolemy's theorem (cf. [[Ptolemeus theorem|Ptolemeus theorem]]): For any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281070.png" /> in the plane of a triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281071.png" /> the inequality
in the plane of a triangle $  ABC $
 
the inequality
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281072.png" /></td> </tr></table>
AB \cdot CP + BC \cdot AP  \geq  AC \cdot BP
 
$$
 
  
holds, with equality if and only if $  P $
+
holds, with equality if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281073.png" /> is on the arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281074.png" /> of the circumcircle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281075.png" /> (in this last case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281076.png" /> is a circle quadrangle).
is on the arc $  CA $
 
of the circumcircle of $  ABC $(
 
in this last case, $  ABCP $
 
is a circle quadrangle).
 
  
Brahmagupta's formula states that for any convex cyclic quadrangle $  ABCD $
+
Brahmagupta's formula states that for any convex cyclic quadrangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281077.png" /> with area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281078.png" />, sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281079.png" /> and semi-perimeter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281080.png" />, the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281081.png" /> holds. In general, for any quadrangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281082.png" />, the area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281083.png" /> satisfies
with area $  ( ABCD) $,  
 
sides $  a, b, c , d $
 
and semi-perimeter $  s = ( a+ b + c + d)/2 $,  
 
the relation $  ( ABCD) = \sqrt {( s- a)( s- b)( s- c)( s- d) } $
 
holds. In general, for any quadrangle $  ABCD $,  
 
the area $  ( ABCD) $
 
satisfies
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281084.png" /></td> </tr></table>
( ABCD)  ^ {2}  = ( s- a)( s- b)( s- c)( s- d) -
 
abcd  \cos  ^ {2}  A+
 
\frac{C}{2}
 
.
 
$$
 
  
 
It follows that among all quadrangles with given side lengths the inscribed quadrangles have maximum area (the cyclic order of the sides is immaterial).
 
It follows that among all quadrangles with given side lengths the inscribed quadrangles have maximum area (the cyclic order of the sides is immaterial).
  
==Regular $  n $-gons.==
+
==Regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281085.png" />-gons.==
A regular $  n $-
+
A regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281086.png" />-gon inscribed in a circle with radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281087.png" /> has perimeter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281088.png" /> and area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281089.png" />; a regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281090.png" />-gon circumscribed about a circle with radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281091.png" /> has perimeter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281092.png" /> and area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072810/p07281093.png" />. See also [[Regular polyhedra|Regular polyhedra]].
gon inscribed in a circle with radius $  R $
 
has perimeter $  2nR  \sin ( \pi /n) $
 
and area $  ( n/2) R  ^ {2}  \sin ( 2 \pi /n) $;  
 
a regular $  n $-
 
gon circumscribed about a circle with radius $  R $
 
has perimeter $  2nR  \mathop{\rm tan} ( \pi /n ) $
 
and area $  nR  ^ {2}  \mathop{\rm tan} ( \pi / n) $.  
 
See also [[Regular polyhedra|Regular polyhedra]].
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1969)  pp. 3–23</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  S.L. Greitzer,  "Geometry revisited" , Random House  (1967)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''II''' , Springer  (1987)  pp. Chapt. 10</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1969)  pp. 3–23</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  S.L. Greitzer,  "Geometry revisited" , Random House  (1967)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''II''' , Springer  (1987)  pp. Chapt. 10</TD></TR></table>

Revision as of 14:52, 7 June 2020

trigonometry in the Euclidean plane.

The elements of a triangle, its sides and its angles ( opposite to , etc.), are related in various ways. In the Euclidean plane the most important relations are the angle sum formula

(angles in radians), and the triangle inequalities

These inequalities are necessary and sufficient for three segments of positive length to form the sides of a triangle.

Another relation is the cosine theorem:

In particular, when , the triangle is right-angled, and the cosine theorem becomes Pythagoras' theorem (cf. Pythagoras theorem)

In such a right-angled triangle,

In a general triangle, further relations are provided by the sine theorem:

where is the radius of the circumcircle of the triangle (cf. Inscribed and circumscribed figures). A corollary of the sine theorem is the tangent formula

With the notation for the semi-perimeter of the triangle, the following half-angle formulas follow from the cosine theorem:

Geometry of the triangle.

Among the many remarkable lines, points and circles connected with any triangle are the circumcircle with centre 0 and radius , the incircle and the three excircles with centres , , , and radii , , , (cf. Inscribed and circumscribed figures), the medians (cf. Median (of a triangle)) , , , with the centroid as their common point, the inner bisectors , , , and the outer bisectors , , , the altitude lines , , with the orthocentre as their common point, the Euler line (cf. Euler straight line) through , and , and the nine-point circle through the midpoints of the sides, the feet of the altitude lines, and the midpoints of the segments connecting the vertices of the triangle to its orthocentre. The nine-point circle has radius , its centre is on the Euler line between and such that , and the nine-point circle touches the incircle and the three excircles (Feuerbach's theorem).

With the notation for the area of the triangle , the following relations are valid:

It follows that, among others,

Very remarkable is Morley's theorem: The points of intersection of the adjacent trisectors of the angles of any triangle form the vertices of an equilateral triangle. Indeed, a direct calculation shows that the sides of Morley's triangle have length

which is symmetric in , and .

The theorems of Ceva and Menelaus.

Let be points on the (possibly extended) sides of a triangle . Then, by the Ceva theorem, , and are congruent if and only if (signed distances) and by the Menelaus theorem, , and are collinear if and only if .

Convex quadrangles.

Ptolemy's theorem (cf. Ptolemeus theorem): For any point in the plane of a triangle the inequality

holds, with equality if and only if is on the arc of the circumcircle of (in this last case, is a circle quadrangle).

Brahmagupta's formula states that for any convex cyclic quadrangle with area , sides and semi-perimeter , the relation holds. In general, for any quadrangle , the area satisfies

It follows that among all quadrangles with given side lengths the inscribed quadrangles have maximum area (the cyclic order of the sides is immaterial).

Regular -gons.

A regular -gon inscribed in a circle with radius has perimeter and area ; a regular -gon circumscribed about a circle with radius has perimeter and area . See also Regular polyhedra.

References

[a1] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) pp. 3–23
[a2] H.S.M. Coxeter, S.L. Greitzer, "Geometry revisited" , Random House (1967)
[a3] M. Berger, "Geometry" , II , Springer (1987) pp. Chapt. 10
How to Cite This Entry:
Plane trigonometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Plane_trigonometry&oldid=48186
This article was adapted from an original article by J. van de Craats (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article