TRIANGLE, in geometry, a figure of three sides and three angles. 'Triangles are either plane or spherical. A plane triangle is contained under three right lines ; and a spherical one is a triangle contained under three arches of great cir cles of the sphere. Triangles are denomi nated, their angles, right, obtuse, and acute. A right-angled triangle is that which has one right angle. An obtuse angled triangle is such as has one ob tuse angle. And an acute-angled tri angle is that which has all its angles acute.
In every triangle the sines of the sides are proportional to the sines of the oppo site angles ; also the sine of all the three angles is equal to two right ones ; and the external angle, made by any side produ ced, is equal to the sum of the two inter nal and opposite angles. Triangles on the same base, and having the same height or place, between the same paral lels, are equal ; also triangles on equal bases, and between the same parallels, are equal. If a perpendicular be let fall upon the base of an oblique-angled triangle, the difference of the squares of the sides is equal to the double rectangle under the base, and the dis tance of the perpendicular from the mid dle of the base. The side of an equilate ral triangle, inscribed in a circle, is in power triple of the radius. The sides of a triangle are cut proportionably, by a line drawn parallel to its base. A whole triangle is to a triangle cut off by a right line drawn parallel to the base, as the rectangle:. under the cut sides is to the rectangle of the two other sides. In a right-angled triangle, a line drawn from the right angle at the top, perpendicular to the hypothenuse, divides the triangle into two other right-angled triangles, which are similar to the first triangle, and to one another. In every right-angled
triangle, the square of the hypothenuse is equal to the sum of the squares of the other two sides ; and, in general, any fi gure described on the hypothenuse, is equal to the sum of two similar figures described upon the two shies. In an isos celes triangle, that is, a triangle having two of its sides equal, it a line be drawn from the vertex to any point in the base, the square of that line together with the rectangle of the segments of the base, is equal to the square of the side. If one angle of a triangle be equal to 120°, the square of the base will be equal to the squares of both sides, together with the rectangle of those sides ; and if those sides be equal to each other, then the square of the base will be equal to three times the square of one side, or equal to twelve times the square of the perpendi cular from the angle upon the base.
If any angle of . a triangle be bisected, the bisecting line will divide the opposite side in the same proportion as the Legs of the angle are to one another. Every tri angle is one half of a parallelogram of the same base and height. The area of any triangle may be had by adding all the three sides together, and taking half the sum, and from that half subtracting each side severally, and multiplying that half sum and the remainder continually into one another, and extracting the squaw root of the product. See TRI GONOMR TRY.