TRIANGLE (ties, three, angulus, a corner), the most simple of geometrical figures, is a figure having three angles; but, oddly enough, it is generally defined by geometers as a figure of three sides, and its property of being three-angled is put in the subordinate position of a necessary consequence. It may be that this arises from Euclid's use of the word tripleuron (three-sided) in the definitions prefixed to his Elements; while trigonon (three-angled) is employed in the work itself.
In plane geometry, a triangle is bounded by three straight lines; and triangles are classed according to the relative length of their sides, into equilateral or equal sided; isosceles, or having two sides equal; and scalene, or unequal-sided, the equality or ine quality of the sides carrying with it the equality or similar inequality (of greater or less) of the angles respectively opposite to these sides, though the satin of inequality of the sides by no means corresponds to that of the angles. Considered with reference to the size of its angles, a triangle is right-angled when one of its angles is a right angle (90°); obtuse angled, when it has one angle greater than a right angle; and acute-angled when it has no angle so great as a right angle; the well-known property, that the sum of the angles of a triangle is equal to two right angles, preventing the possibility of more time one of them being as great as a right angle. For the relations between the sides and angles of a triangle, see TRIGONOMETRY. The triangle being the fundamental figure of plane geometry, through which the properties of all other figures have been arrived at, the investigation of its properties has always been held to be of primary importance. Of
the immense number of results obtained by investigation, we can notice only two or three iu this place. The lines joining the angles of a triangle with the points of bisec tion of the opposite sides, intersect at the same point, as also do the perpendiculars from the angles on the opposite sides, the lines bisecting the angles, and the perpendiculars from the middle points of the sides. The point of intersection of the first series of lines is the center of gravity of the triangle; those of the third and fourth series are the cen ters of two circles, the former of which touches the sides internally, and the latter passes through its three angular points. Another remarkable property of triangles, known as Napoleon's problem, is as follows: if on any triangle three equilateral triangles be described, and the centers of gravity of these three be joined, the triangle thus formed. is equilateral, and has its center of gravity coincident with that of the original triangle. See also TRIGONOMETRY and livrommiusE. The area Of a triangle is half of that of a parallelogram which has the same base and altitude, and is thus equal to half the product of the base into the altitude; it may also be expressed by the formula 4/S(S--a)(S—b)(S—c), where a, b, c. are the lengths of the sides, anhi S is half their F1171. In the geometry of the sphere, a triangle is a figure bounded by three arcs of circles.