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''trigonometry in the Euclidean plane.''

The elements of a triangle, its sides  $  a, b, c $
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

$$ 
A + B + C  =  \pi
$$

(angles in radians), and the triangle inequalities

$$ 
a + b  >  c ,\ \ 
b + c  >  a ,\ \ 
c + a  >  b .
$$

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

Another relation is the [[Cosine theorem|cosine theorem]]:

$$ 
c  ^ {2}  =  a  ^ {2} + b  ^ {2} - 2ab  \cos  C.
$$

In particular, when  $  C = \pi /2 $, 
the triangle is right-angled, and the cosine theorem becomes Pythagoras' theorem (cf. [[Pythagoras theorem|Pythagoras theorem]])

$$ 
c  ^ {2}  =  a  ^ {2} + b  ^ {2} .
$$

In such a right-angled triangle,

$$

In a general triangle, further relations are provided by the [[Sine theorem|sine theorem]]:

$$ 

\frac{a}{\sin  A }
  =  
\frac{b}{\sin  B }
  = \ 

\frac{c}{\sin  C }
  =  2R ,
$$

where  $  R $
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

$$ 
a- 
\frac{b}{a+}
 b  = \ 

\frac{ \mathop{\rm tan} [( A- B)/2] }{ \mathop{\rm tan} [( A+ B)/2] }
  = \ 
 \mathop{\rm tan}  A- 
\frac{B}{2}
   \mathop{\rm cotan}  
\frac{C}{2}
 .
$$

With the notation  $  s = ( a+ b+ c)/2 $
for the semi-perimeter of the triangle, the following half-angle formulas follow from the cosine theorem:

$$ 
\cos  ^ {2}  
\frac{A}{2}
  =  s( s- 
\frac{a)}{bc}
 ,
$$

$$ 
\sin  ^ {2}  
\frac{A}{2}
  =  ( s- b)( s- 
\frac{c)}{bc}
 ,
$$

$$ 
 \mathop{\rm tan}  ^ {2}  
\frac{A}{2}
  =  ( s- b)( s- 
\frac{c)}{s(}
 s- a) .
$$

==Geometry of the triangle.==
Among the many remarkable lines, points and circles connected with any triangle are the circumcircle with centre 0 and radius  $  R $, 
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) $
for the area of the triangle  $  ABC $, 
the following relations are valid:

$$ 
( ABC)  =  
\frac{1}{2}
 ah _ {a}  =  
\frac{1}{2}
 bc  \sin  A  = \ 

\frac{abc}{4R\ }
 =
$$

$$ 
= \ 
r \cdot s  =  r _ {a} ( s- a)  =  \sqrt {s( s- a)( s- b)( s- c) } .
$$

It follows that, among others,

$$ 
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

$$ 
8R  \sin  
\frac{A}{3}
  \sin  
\frac{B}{3}
  \sin  
\frac{C}{3}
 ,
$$

which is symmetric in  $  A $, 
$  B $
and  $  C $.

==The theorems of Ceva and Menelaus.==
Let  $  X, Y, Z $
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.==
Ptolemy's theorem (cf. [[Ptolemeus theorem|Ptolemeus theorem]]): For any point  $  P $
in the plane of a triangle  $  ABC $
the inequality

$$ 
AB \cdot CP + BC \cdot AP  \geq   AC \cdot BP
$$

holds, with equality if and only if  $  P $
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 $
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

$$ 
( ABCD)  ^ {2}  =  ( s- a)( s- b)( s- c)( s- d) -
abcd  \cos  ^ {2}  A+ 
\frac{C}{2}
 .
$$

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

==Regular  $  n $-gons.==
A regular  $  n $-
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====
<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>