**TRIANGLE,** in geometry, a figure enclosed by three lines; if the lines be straight the figure is called a plane triangle; but
if the figure be enclosed by lines on the surface of a sphere it is a
spherical triangle. The latter are treated in Trigonometry;
here we summarize the more important properties of plane
triangles. In a plane triangle any one of the angular points can
be regarded as the vertex; and the opposite side is called the base.
The three sides and angles constitute the six elements of a
triangle; it is customary to denote the angular points by capital
letters and refer to the angles by these symbols; the sides are
usually denoted by the lower case letter corresponding to that of
the opposite angular point. Triangles can be classified according
to the relative sizes of the sides or angles. An equilateral triangle
has its three sides equal; an isosceles triangle has only two
sides equal; whilst a scalene triangle has all its sides unequal.
Also a right-angled triangle has one angle a right angle, the side
opposite this angle being called the hypotenuse; an obtuse angled
triangle has one angle obtuse, or greater than a right
angle; an acute-angled triangle has three acute angles, *i.e.*
angles less than right angles. The triangle takes a prominent
place in book i. of Euclid; whilst the relation of the triangle
to certain circles is treated in book iv. (See Geometry:
§ *Euclidean*.)

The following is a summary of the Euclidean results. The angles
at the base of an isosceles triangle are equal and conversely; hence
it follows that an equilateral triangle is also equiangular and conversely
(i. 5, 6). If one side of a triangle be produced then the
exterior angle is greater than either of the two interior opposite
angles (i. 16), and equal to their sum (i. 32); hence the sum of
the three interior angles equals two right angles. (In i. 17 it is
shown that any two angles are less than two right angles.) The
greatest angle in a triangle is opposite the greatest side (i. 18, 19).
On the identical equality of triangles Euclid proves that two triangles
are equal in all respects when the following parts are equal
each to each (a) two sides, and the included angle (i. 4), three sides
(i. 8, cor.), two angles and the adjacent side, and two angles and the
side opposite one of them (i. 26). The mensuration is next treated.
Triangles on the same base and between the same parallels, *i.e.*
having the same altitude, are equal in area (i. 37); similarly triangles
on equal bases and between the same parallels are equal in area
(i. 38). If a parallelogram and triangle be on the same base and
between the same parallels then the area of the parallelogram is
double that of the triangle (i. 41). These propositions lea to the
result that the area of a triangle is one half the product of the base
into the altitude. The penultimate proposition (i. 47) establishes
the beautiful theorem, named after Pythagoras, that in a right angled
triangle the square on the hypotenuse equals the sum of
the squares on the other two sides. Two important propositions
occur in book ii. viz. 12 and 13; these may be stated in the following
forms: If ABC is an obtuse-angled triangle with the obtuse
angle at C and a perpendicular be drawn from the angular point A
cutting the base BC produced in D, then AB^{2} (*i.e.* square on the side
subtending the obtuse angle) = BC^{2} + CA^{2} + 2BC·CD (ii. 12);
in any triangle (with the sarne construction but with the side AC
subtending an acute angle B, we have AC^{2} = AB^{2} + BC^{2} − 2CB·BD
(see Trigonometry).

Book iv. deals with the circles of a triangle. To inscribe a circle in
a given triangle is treated in iv. 4; to circumscribe a circle to a given
triangle in iv. 5. The centre of the first circle is the intersection of
the bisectors of the interior angles; if the meet of the bisectors of
two exterior angles be taken, a circle can be drawn with this point
as centre to touch two sides produced and the third side; three such
circles are possible and are called the escribed circles. The centre
of the circum circle is the intersection of the perpendiculars from the
middle-points of the sides. Concerning the circum circle we observe
that the feet of the perpendiculars drawn from any point on its
circumference to the sides are collinear, the line being called Simson's
line. We may here notice that the perpendiculars from the vertices
of a triangle to the opposite sides are concurrent; their meet is called
the orthocentre, and the triangle obtained by joining the feet of
the perpendiculars is called the pedal triangle. Also the lines
joining the middle point of the sides to the opposite vertices, or
medians, are concurrent in the centroid or centre of gravity of the
triangle. There are several other circles, points and lines of
interest in connexion with the triangle. The most important is
the “nine point circle,” so called because it passes through (*a*) the
middle points of the sides; (*b*) the feet of the perpendiculars from
the vertices to the opposite sides; and (*c*) the middle points of the
lines joining the or tho centre to the angular points. This circle
touches the inscribed and escribed circles. For the Brocard points
and circle, Tucker’s circles—with the particular forms cosine
circle, triplicate ratio (T.R.) circle, Taylor's circle, McCay's circles,
&c., see W. J. M’Clelland, *Geometry of the Circle*; or Casey, *Sequel*
*to Euclid*.