The Euclid's Elements of Geometry

Geometry is the study of figures. Euclid's Elements provides themost fundamental way of learning geometry geometrically.

Book VI: Similar figures and proportions in geometry

Definitions

1. Similar rectilinear figures are such as have their angles severally equal and the sides about the equal angles proportional.
2. Two figures are reciprocally related when the sides about corresponding angles are reciprocally proportional.
3. A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.
4. The height of any figure is the perpendicular drawn from the vertex to the base.

Propositions

1.  Triangles and parallelograms which are under the same height are to one another as their bases.
2. If a straight line is drawn parallel to one of the sides of a triangle, then it cuts the sides of the triangle proportionally; and, if the sides of the triangle are cut proportionally, then the line joining the points of section is parallel to the remaining side of the triangle.
3. If an angle of a triangle is bisected by a straight line cutting the base, then the segments of the base have the same ratio as the remaining sides of the triangle; and, if segments of the base have the same ratio as the remaining sides of the triangle, then the straight line joining the vertex to the point of section bisects the angle of the triangle.
4. In equiangular triangles the sides about the equal angles are proportional where the corresponding sides are opposite the equal angles.
5. If two triangles have their sides proportional, then the triangles are equiangular with the equal angles opposite the corresponding sides.
6. If two triangles have one angle equal to one angle and the sides about the equal angles proportional, then the triangles are equiangular and have those angles equal opposite the corresponding sides.
7. If two triangles have one angle equal to one angle, the sides about other angles proportional, and the remaining angles either both less or both not less than a right angle, then the triangles are equiangular and have those angles equal the sides about which are proportional.
8. If in a right-angled triangle a perpendicular is drawn from the right angle to the base, then the triangles adjoining the perpendicular are similar both to the whole and to one another.
   Corollary: If in a right-angled triangle a perpendicular is drawn from the rightangle to the base, then the straight line so drawn is a mean proportionalbetween the segments of the base.
9. To cut off a prescribed part from a given straight line.
10. To cut a given uncut straight line similarly to a given cut straight line.
11. To find a third proportional to two given straight lines.
12. To find a fourth proportional to three given straight lines.
13. To find a mean proportional to two given straight lines.
14. In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal.
15. In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional; and those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal.
16. If four straight lines are proportional, then the rectangle contained by the extremes equals the rectangle contained by the means; and, if the rectangle contained by the extremes equals the rectangle contained by the means, then the four straight lines are proportional.
17. If three straight lines are proportional, then the rectangle contained by the extremes equals the square on the mean; and, if the rectangle contained by the extremes equals the square on the mean, then the three straight lines are proportional.
18. To describe a rectilinear figure similar and similarly situated to a given rectilinear figure on a given straight line.
19. Similar triangles are to one another in the duplicate ratio of the corresponding sides.
    Corollary: If three straight lines are proportional, then the first is to thethird as the figure described on the first is to that which is similar andsimilarly described on the second.
20. Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding side.
    Corollary: Similar rectilinear figures are to one another in the duplicate ratioof the corresponding sides.
21. Figures which are similar to the same rectilinear figure are also similar to one another.
22. If four straight lines are proportional, then the rectilinear figures similar and similarly described upon them are also proportional; and, if the rectilinear figures similar and similarly described upon them are proportional, then the straight lines are themselves also proportional.
23. Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.
24. In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another.
25. To construct a figure similar to one given rectilinear figure and equal to another.
26. If from a parallelogram there is taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, then it is about the same diameter with the whole.
27. Of all the parallelograms applied to the same straight line falling short by parallelogrammic figures similar and similarly situated to that described on the half of the straight line, that parallelogram is greatest which is applied to the half of the straight line and is similar to the difference.
28. To apply a parallelogram equal to a given rectilinear figure to a given straight line but falling short by a parallelogram similar to a given one; thus the given rectilinear figure must not be greater than the parallelogram described on the half of the straight line and similar to the given parallelogram.
29. To apply a parallelogram equal to a given rectilinear figure to a given straight line but exceeding it by a parallelogram similar to a given one.
30. To cut a given finite straight line in extreme and mean ratio.
31. In right-angled triangles the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle.
32. If two triangles having two sides proportional to two sides are placed together at one angle so that their corresponding sides are also parallel, then the remaining sides of the triangles are in a straight line.
33. Angles in equal circles have the same ratio as the circumferences on which they stand whether they stand at the centers or at the circumferences.