41. Definition. When a given straight line is divided into two segments such that one of the segments is a mean proportional between the given line and the other segment, it is said to be divided in extreme and mean ratio.

Thus, AB is divided in extreme and mean ratio at C, if AB AC = AC : CB.

A

с

B

PROPOSITION XVII.-PROBLEM.

42. To divide a given straight line in extreme and mean ratio. Let AB be the given straight line. At B erect the perpendicular BO equal to one-half of AB. With the centre and radius OB, describe a circumference, and through A and O draw AO cutting the circumference first in D and a second

time in D'. Upon AB lay off AC=AD. Then AB is divided at C in extreme and mean ratio.

D

B

For we have (Proposition XII., Corollary)

or, since DD'=20B AB, and therefore AD'ABAD

=

AC: AB= CB: AC,

and, by inversion (7),

AB: AC AC: CB;

=

that is, AB is divided at C in extreme and mean ratio.

PROPOSITION XVIII.-PROBLEM.

43. On a given straight line, to construct a polygon similar to

a given polygon.

Let it be required to construct upon A'B' a polygon similar to ABCDEF.

A'B'C' will be similar to ABC (Proposition III.). In the same manner construct the triangle A'D'C' similar to ADC, A'E'D' similar to AED, and A'E'F' similar to AEF. Then A'B'C'D'E'F' is the required polygon (Proposition VI.).

EXERCISES ON BOOK III.

THEOREMS.

1. If two straight lines are intersected by any number of parallel lines, the corresponding segments of the two lines are proportional. (v. Proposition I.)

2. The diagonals of a trapezoid divide each other into segments which are proportional.

3. In a triangle any two sides are reciprocally proportional to the perpendiculars let fall upon them from the opposite vertices. 4. The perpendiculars from two vertices of a triangle upon the opposite sides divide each other into segments which are reciprocally proportional.

5. If the three sides of a triangle are respectively perpendicular to the three sides of a second triangle, the triangles are similar. 6. If ABC and A'BC are two triangles having a common base BC and their vertices in a line AA' parallel to the base, and if any parallel to the base cuts the sides AB and AC in D and E, and the sides A'B and A'C in D' and E', then DE

D'E' (Proposition III.).

7. If two sides of a triangle are divided proportionally, the straight lines drawn from corresponding points of section to the opposite angles intersect on the line joining the vertex of the third angle and the middle of the third side.

Suggestion. Draw the line ADE through the intersection of B'C and BC'. B'E'D and CED are

E

D

C

B

8. The difference of the squares of two sides of any triangle is equal to the difference of the squares of the projections of these sides on the third side (Proposition X.).

9. If from any point in the plane of a polygon perpendiculars are drawn to all the sides, the two sums of the squares of the alternate segments of the sides are equal.

10. If through a point P in the circumference of a circle two chords are drawn, the chords and the segments cut from them by a line parallel to the tangent at P are reciprocally proportional.

Suggestion. Prove PAB and Pba similar.

11. If three circles intersect, their three common chords pass through the same point. (v. Proposition XI.)

12. If two tangents are drawn to a circle at the extremities of a diameter, the portion of any third tangent intercepted between them is divided at its point of contact into segments whose product is equal to the square of the radius.

Suggestion. Prove AOB a right triangle.

T

13. The perpendicular from any point of a circumference upon a chord is a mean proportional between the perpendiculars from the same point upon the tangents drawn at the extremities of the chord.

Suggestion. PBD and PAE are similar;

14. If two circles touch each other, secants drawn through their point of contact and terminating in the two circumferences are divided proportionally at the point of contact. (v. II., 54, Exercise 2.)

15. If two circles are tangent externally, the portion of their common tangent included between the points of contact is a mean proportional between the diameters of the circles.

Suggestion. Show that OBO' is a right triangle.

T

B

༡

16. If a fixed circumference is cut by any circumference which passes through two fixed points, the common chord passes through a fixed point.

Suggestion. PA.PB = PC. PD=PT2, by Proposition XII. and Corollary. Join P with C', and show that PC' will cut both circles at the same distance from P, and will be their common chord.

D'

D

T

BP

LOCI.

17. From a fixed point O, a straight line OA is drawn to any point in a given straight line MN, and divided at P in a given ratio m:n (i.e, so that OP: PA: = m:n); find the locus of P. (v. Proposition II.)

18. From a fixed point O, a straight line OA is drawn to any point in a given circumference, and divided at P in a given ratio; find the locus of P.

Suggestion. PC' is a fixed length.

19. Find the locus of a point whose distances from two given straight lines are in a given ratio. (The locus consists of two straight lines.)