ratios of the including sides of the first to the in

cluding sides of the second.

COR. If two triangles or parallelograms have one angle of the one supplementary to one angle of the other, their areas have to one another the ratio compounded of the ratios of the including sides of the first to the including sides of the second.

COR. The ratio compounded of two ratios between straight lines is the same as the ratio of the rectangle contained by the antecedents to the rectangle contained by the consequents.

THEOR. 14. Triangles and parallelograms have to one another the ratio compounded of the ratios of their bases and

of their altitudes.

THEOR. 15. Similar triangles are to one another in the duplicate ratio of their homologous sides.

THEOR. 16. The areas of similar rectilineal figures are to one another in the duplicate ratio of their homologous

sides.

COR. 1. Similar rectilineal figures are to one another as the squares described on their homologous sides.

COR. 2. If four straight lines are proportional and a pair of

similar rectilineal figures are similarly described on the first and second, and also a pair on the third and fourth, these figures are proportional; and conversely, if a rectilineal figure on the first of four straight lines is to the similar and similarly described figure on the second as a rectilineal figure on the third is to the similar and similarly described figure on the fourth, the four straight lines are proportional. THEOR. 17. In any right-angled triangle, any rectilineal figure

58

A SYLLABUS OF PLANE GEOMETRY.

Loci.

described on the hypotenuse is equal to the sum of two similar and similarly described figures on the sides.

SECTION 3.

LOCI AND PROBLEMS.

i. The locus of a point whose distances from two fixed straight lines are in a constant ratio is a pair of straight lines, passing through the point of intersection of the given lines, if they intersect, and parallel to them, if the lines are parallel.

ii. The locus of a point whose distances from two fixed points are in a constant ratio (not one of equality) is

a circle.

PROB. I. To divide a straight line similarly to a given divided straight line.

PROB. 2. To divide a straight line internally or externally in a given ratio.

PROB. 3. From a given straight line to cut off any part

required.

PROB. 4. To find a fourth proportional to three given straight

lines.

PROB. 5. To find a mean proportional between two given. straight lines.

PROB. 6. On a straight line to describe a rectilineal figure

similar to a given rectilineal figure.

PROB. 7. To describe a rectilineal figure equal to one and similar to another given rectilineal figure.

CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS,