BOOK III. ON PLANE FIGURES. CHAPTER I. ON RECTILINEAR PLANE FIGURES. I. SECTION I. ON POLYGONS GENERALLY. DEFINITIONS. A plane figure is a determinate portion of a plane. It has been shown [Gen. Def. 5] that every portion of a surface is limited by lines; consequently, a plane figure is limited by one or more lines. A plane figure is rectilinear or curvilinear, according to the kind of lines by which it is limited. Whence 2. A rectilinear plane figure, or polygon, is limited by straight lines, called its sides; two sides having one point in common are adjacent to each other. The perimeter of a polygon is the sum of all its sides; the semiperimeter is half that sum. It is evident that no plane figure can be formed by less than three straight lines. 3. The angles formed by adjacent sides of a polygon are the interior angles, or simply the angles, of the polygon; those formed by each side with the prolongation of its adjacent, are the exterior angles; their vertices are the angular points, or vertices of the polygon. Adjacent vertices of a polygon are the extremities of the same side. 4. A diagonal line, or simply diagonal, joins two vertices not adjacent to each other; the diagonals adjacent to one side are terminating at its extremities. The primary diagonals subtend the interior angles; the secondary diagonals subtend the angles formed by the primary diagonals and the sides. 5. A convex polygon has only projecting angles; a concave polygon has one or more receding angles.1 Particular names have been given to polygons of a certain number of sides. Thus 6. A polygon of three sides is a triangle, or trigon; 1 Whenever in The Elements a polygon is spoken of, without anything further being said, a convex polygon is always meant. 7. An equilateral polygon is that of which all the sides are equal to each other; an equiangular polygon is that of which all the angles are equal to each other. THEOREM 1. In every polygon there are as many vertices as there are sides. For if the sides be numbered, the first will begin at the first vertex and meet the second side; the second will begin at the second vertex and meet the third side; and so on to the last side, which will begin at the last vertex and meet the first side at the first vertex. COROLLARY. The straight lines joining a point within a polygon to all its angular points, divide the polygon into as many triangles as there are sides. THEOREM 2. If all the diagonals be drawn from one vertex of a polygon, there are three diagonals less than there are sides. Each vertex may be joined to all the others by as many straight lines as there are other vertices [Post. V], that is, one less than there are sides: but two of these straight lines are sides of the polygon, and the others, diagonals; therefore there are three diagonals less than there are sides, W. W. T. B. D. COROLLARY. The diagonals drawn from one vertex, divide the polygon into as many triangles as there are sides but two. THEOREM 3. (Eucl. I. 32.) The three angles of every triangle are together equal to two right angles. |