TRIANGLE

(Lat. triangulum.) A three-sided figure, which consequently has also three angles. Triangles are rectilineal, spherical, or curvilineal. Rectilineal or plane triangles are bounded by straight lines; spherical triangles are formed on the surface of a sphere by the intersection of the planes of three great circles; curvilineal triangles are those which are bounded by three curves of any kind whatever. Triangles receive other distinctive denominations from the relation of their sides and angles. Plane triangles are denominated equilateral when their three sides are all equal; isosceles when two only are equal; and scalene when all three are unequal. They are called right-angled when one of the angles is a right angle; oblique-angled when one of the angles is more than a right angle; and acute-angled when each of the angles is le than a right angle; and triangles are said to be similar when their angles are respectively equal each to each. The following are some of the principal properties of plane triangles; 1. The greater side is opposite the greater angle, and rice vers a. 2. The sum of any two sides is greater, and the differ- "ence of any two le than the third side. 3. The sum of the three angles is equal to two right angles;and if one of the sides be produced, the exterior angle is equal to both the interior and opposite ones. 4. If A E and C D be perpendiculars from the angles A and C on the opposite sides A B and B C, then, generally, AC2 = AB-AD + CBC E; and, consequently, if the angle B is a right angle, A C-! = A B2+ B C2. 5. Lines which bisect the angles of a plane triangle will pa through the centre O of the inscribed circle; perpendiculars from the middle of the sides pa through the centre of the circumscribing circle; and lines drawn from the angles to the middle of the opposite sides pa through the centre of gravity. 6. Triangles are to each other in the compound ratio of their bases and altitudes. The area of a plane triangle is found by multiplying the length of one of the sides into that of the perpendicular let fall upon it from the opposite angle; but there are various other expre ions for the area, for which see Trigonometry. Triangle. In Astronomy, one of the forty-eight constellations of Hipparch us, situated in the northern hemisphere. The same name is also given to one of the twelve southern constellations formed by Bayer. There is also the Little Triangle, added by Hevelius, near the first named. See Constellation. Triangle. In Music, a small steel triangular musical Instrument of percu ion, open at one of its angles. It is iet in vibration by being struck with a short bar of the same metal as that whereof the instrument is composed. TRIANGLE, [s. 1280]