MST Topics in the History of Mathematics

Size: px

Start display at page:

Download "MST Topics in the History of Mathematics"

Annabella Rachel Gregory
5 years ago
Views:

1 MST Topics in the History of Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Summer B: Euclid s Book VI (Similar triangles)

2 Euclide VI.1,2 VI.1. Triangles and parallelograms which are under the same height are to one another as their bases. VI.2. If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally; and, if the sides of the triangle be cut proportionally, the line joining the points of section will be parallel to the remaining side of the triangle. 1

3 Euclid VI.3: Angle bisector theorem If an angle of a triangle is bisected and the straight line cutting the angle cuts the base also, the segments of the base will have the same ratio of the remaining sides of the triangle; and if the segments of the base have the same ratio as the remaining sides of the triangle, the straight line joined from the vertex to the point of the section will bisect the angle of the triangle. B D A C E 2

4 External angle bisector theorem This is also true for the bisector of the external angle: If E is a point on the extension of the side BC, AE bisects an external angle of BAC if and only if BE : EC = AB : AC. A B D C Figure 2: E Corollary. Let B and C be given points. The locus of a point P whose distances from B and C are in a constant ratio (not equal to 1) is a circle. 3

5 Euclid VI.4,5 VI.4. In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles. VI.5. If two triangles have their sides proportional, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend. 4

6 Euclid VI.6,7 VI.6. If two triangles have one angle equal to one angle and the sides about the equal angles proportional, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend. VI.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, the triangles will be equiangular and will have those angles equal, the sides about which are proportional. 5

7 Euclid VI.8 If in a right-angled triangle a perpendicular be drawn from the right angle to the base, the triangles adjoining the perpendicular are similar both to the whole and to one another. B A D Figure 3: Euclid VI.8 Corollary. From this it is clear that, if in a right-angled triangle a perpendicular be drawn from the right angle to the base, the straight line so drawn is a mean proportional between the segments of the base. C 6

8 Euclid VI.9 12 VI.9. From a given straight line to cut off a prescribed part. VI.10. To cut a given uncut straight line similarly to a given cut straight line. VI.11. To two given straight lines to find a third proportional. VI.12. To three given straight lines to find a fourth proportional. 7

9 Euclid VI.13 To two given straight lines to find a mean proportional. A B D C 8

10 Proposition 0.1 (VI.19). Similar triangles are to one another in the duplicate ratio of the corresponding sides. Proposition 0.2 (VI.31). In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle. Proposition 0.3 (VI.32). If two triangles having two sides proportional to two sides be placed together at one angle so that their corresponding sides are also parallel, the remaining sides of the triangles will be in a straight line. D A B C E Figure 4: Euclid VI.32 Proposition 0.4 (VI.33). In equal circles angles have the same ratio as the circumferences on which they stand, whether they stand at the centers or at the circumferences. 9

MST Topics in History of Mathematics

MST Topics in History of Mathematics Euclid s Elements and the Works of rchimedes Paul Yiu Department of Mathematics Florida tlantic University Summer 2014 June 30 2.6 ngle properties 11 2.6 ngle properties