Triangle

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 hypothenuse; an obtuseangled 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 lead 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 rightangled triangle the square on the hypothenuse 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 (i.e. square on the side subtending the obtuse angle) = BC + CA' 2BC'CD (ii. 12); in any triangle (with the same construction but with the side AC subtending an acute angle B, we have AC = AB + BC - 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. Th 2 centre of the circum circle is the intersection of the perpendiculars ' m the middle points of the sides. Concerning the circum circle w ..serve that the feet of the perpendiculars drawn from any poin 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 orthocentre 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.