PROB. 7. To construct a triangle having its sides equal to three given straight lines, any two of which are greater than the third. PROB. 8. To construct a triangle, having given two sides and the angle between them. PROB. 9. To construct a triangle, having given two sides and an angle opposite to one of them. PROB. 10. To construct a triangle, having given two angles and the side that forms their common arm. PROB. II. To construct a triangle, having given two angles and a side opposite to one of them. I. 2. If any and every point on a line or group of lines (straight or curved), and no other point, satisfies an assigned condition, that line or group of lines is called the locus of the point satisfying that condition. In order that a line or group of linės A may be properly termed the locus of a point satisfying an assigned condition X, it is necessary and sufficient to demonstrate the two following associated Theorems : If a point is on A, it satisfies X. If a point is not on A, it does not satisfy X. It may sometimes be more convenient to demonstrate the contrapositive of either of these Theorems. i. The locus of a point at a given distance from a given point is the circumference of a circle having a radius equal to the given distance and its centre at the given point. 20 . A SYLLABUS OF PLANE GEOMETRY. ii. The locus of a point at a given distance from a given straight line is the pair of straight lines parallel to the given line, at the given distance from it and on opposite sides of it. iii. The locus of a point equidistant from two given points is the straight line that bisects, at right angles, the line joining the given points. iv. The locus of a point equidistant from two intersecting straight lines is the pair of lines, at right angles to one another, which bisect the angles made by the given lines. 3. Intersection of Loci. If A is the locus of a point satisfying the condition X, and B the locus of a point satisfying the condition Y ; then the intersections of A and B, and these points only, satisfy both the conditions X and Y. i. There is one and only one point in a plane which is equi distant from three given points not in the same straight line. ii. There are four and only four points in a plane each of which is equidistant from three given straight lines that intersect one another but not in the same point. EQUALITY OF AREAS. SECTION 1. THEOREMS. DEF. I. The altitude of a parallelogram with reference to a given side as base is the perpendicular distance between the base and the opposite side. DEF. 2. The altitude of a triangle with reference to a given side as base is the perpendicular distance between the base and the opposite vertex. OBS. It follows from the General Axioms (d) and (e) (page 3), as an extension of the Geometrical Axiom 1 (page 10), that magnitudes which are either the sum or the difference of identically equal magnitudes are equal, although they may not be identically equal. THEOR. I. Parallelograms on the same base and between the same parallels are equal. COR. I. The area of a parallelogram is equal to the area of a rectangle, whose base and altitude are equal to those of the parallelogram. Cor. 2. Parallelograms on equal bases and of equal altitude are equal; and of parallelograms of equal altitudes, that is the greater which has the greater base; and also of parallelograms on equal bases, that is the greater which has the greater altitude. * Book III. (with the exception of its last Section) is independent of Book II., and may be studied immediately after Book I. 22 A SYLLABUS OF THEOR. 2. The area of a triangle is half the area of a rectangle whose base and altitude are equal to those of the triangle. COR. 1. Triangles on the same or equal bases and of equal altitude are equal. COR. 2. Equal triangles on the same or equal bases have equal altitudes. COR. 3. If two equal triangles stand on the same base and on the same side of it, or on equal bases in the same straight line and on the same side of that straight line, the line joining their vertices is parallel to the base or to that straight line. THE R. 3. The area of a trapezium is equal to the area of a rectangle whose base is half the sum of the two parallel sides, and whose altitude is the perpendicular distance between them. DEF. 3. The straight lines drawn through any point in a diagonal of a parallelogram parallel to the sides divide it into four parallelograms, of which the two whose diagonals are upon the given diagonal are called parallelograms about that diagonal, and the other two are called the complements of the parallelograms about the diagonal. THEOR. 4. The complements of parallelograms about the diagonal of any parallelogram are equal to one another. DEF. 4. All rectangles being identically equal which have two adjoining sides equal to two given straight lines, given straight line is spoken of as the square on that line. DEF. 5. A point in a straight line is said to divide it internally, or, simply, to divide it; and, by analogy, a point in the line produced is said to divide it externally; and, in either case, the distances of the point from the extremities of the line are called its segments. OBs. A straight line is equal to the sum or difference of its segments according as it is divided internally or externally. THEOR. 5. The rectangle contained by two given lines is equal to the sum of the rectangles contained by one of them and the several parts into which the other is divided. Cor. 1. If a straight line is divided into two parts, the rect angle contained by the whole line and one of the parts is equal to the sum of the square on that part and the rectangle contained by the two parts. COR. 2. If a straight line is divided into two parts the square on the whole line is equal to the sum of the rectangles contained by the whole line and each of the parts. THEOR. 6. The square on the sum of two lines is greater than the sum of the squares on those lines by twice the rectangle contained by them. THEOR. 7. The square on the difference of two lines is less than the sum of the squares on those lines by twice the rectangle contained by them. THEOR. 8. The difference of the squares on two lines is equal to the rectangle contained by the sum and difference of the lines. THEOR. 9. In any right-angled triangle the square on the |