2. If a transversal is drawn to cut the sides, or the sides produced, of a triangle, the product of three alternate segments taken in order is equal to the product of the other three segments.

Let ABC be a triangle, and let a transversal meet the sides BC, CA, AB, or these sides produced, at D, E, F:

then shall BD.CE.AF=DC. EA . FB.

Draw AH par1 to BC, meeting the transversal at H.

Q. E. D.

NOTE. In this theorem the transversal must either meet two sides and the third side produced, as in Fig. 1; or all three sides produced, as in Fig. 2.

The converse of this Theorem may be proved indirectly:

If three points are taken in two sides of a triangle and the third side produced, or in all three sides produced, so that the product of three alternate segments taken in order is equal to the product of the other three segments, the three points are collinear.

The propositions given on pages 103-106 relating to the concurrence of straight lines in a triangle, may be proved by the method of transversals, and in addition to these the following important theorems may be established.

DEFINITIONS.

(i) If two triangles are such that three straight lines joining corresponding vertices are concurrent, they are said to be copolar.

(ii) If two triangles are such that the points of intersection of corresponding sides are collinear, they are said to be co-axial.

THEOREMS TO BE PROVED BY TRANSVERSALS.

1. The straight lines which join the vertices of a triangle to the points of contact of the inscribed circle (or any of the three escribed circles) are concurrent.

2.

The middle points of the diagonals of a complete quadrilateral are collinear.

3. Co-polar triangles are also co-axial; and conversely co-axial triangles are also co-polar.

4. The six centres of similitude of three circles lie three by three on four straight lines.

MISCELLANEOUS EXAMPLES ON BOOK VI.

1. Through D, any point in the base of a triangle ABC, straight lines DE, DF are drawn parallel to the sides AB, AC, and meeting the sides at E, F: shew that the triangle AEF is a mean proportional between the triangles FBD, EDC.

2. If two triangles have one angle of the one equal to one angle of the other, and a second angle of the one supplementary to a second angle of the other, then the sides about the third angles are proportional.

3. AE bisects the vertical angle of the triangle ABC and meets the base in E; shew that if circles are described about the triangles ABE, ACE, the diameters of these circles are to each other in the same ratio as the segments of the base.

4. Through a fixed point O draw a straight line so that the parts intercepted between O and the perpendiculars drawn to the straight line from two other fixed points may have a given ratio.

5. The angle A of a triangle ABC is bisected by AD meeting BC in D, and AX is the median bisecting BC: shew that XD has the same ratio to XB as the difference of the sides has to their sum.

6. AD and AE bisect the vertical angle of a triangle internally and externally, meeting the base in D and E; shew that if O is the middle point of BC, then OB is a mean proportional between OD and OE.

7. P and Q are fixed points; AB and CD are fixed parallel straight lines; any straight line is drawn from P to meet AB at M, and a straight line is drawn from Q parallel to PM meeting CD at N: shew that the ratio of PM to QN is constant, and thence shew that the straight line through M and N passes through a fixed point.

8. C is the middle point of an arc of a circle whose chord is AB; D is any point in the conjugate arc: shew that

AD + DB: DC :: AB : AC.

9. In the triangle ABC the side AC is double of BC. If CD, CE bisect the angle ACB internally and externally meeting AB in D and E, shew that the areas of the triangles CBD, ACD, ABC, CDE are as 1, 2, 3, 4.

10. AB, AC are two chords of a circle; a line parallel to the tangent at A cuts AB, AC in D and E respectively: shew that the rectangle AB, AD is equal to the rectangle AC, AE.

11. If from any point on the hypotenuse of a right-angled triangle perpendiculars are drawn to the two sides, the rectangle contained by the segments of the hypotenuse will be equal to the sum of the rectangles contained by the segments of the sides.

12.

D is a point in the side AC of the triangle ABC, and E is a point in AB. If BD, CE divide each other into parts in the ratio 4: 1, then D, E divide CA, BA in the ratio 3: 1.

13. If the perpendiculars from two fixed points on a straight line passing between them be in a given ratio, the straight line must pass through a third fixed point.

14. PA, PB are two tangents to a circle; PCD any chord through P: shew that the rectangle contained by one pair of opposite sides of the quadrilateral ACBD is equal to the rectangle contained by the other pair.

15. A, B, C are any three points on a circle, and the tangent at A meets BC produced in D: shew that the diameters of the circles circumscribed about ABD, ACD are as AD to CD.

16. AB, CD are two diameters of the circle ADBC at right angles to each other, and EF is any chord; CE, CF are drawn meeting AB produced in G and H : prove that the rect. CE, HG the rect. EF, CH.

17. From the vertex A of any triangle ABC draw a line meeting BC produced in D so that AD may be a mean proportional between the segments of the base.

18. Two circles touch internally at O; AB a chord of the larger circle touches the smaller in C which is cut by the lines OA, OB in the points P, Q: shew that OP: OQ :: AC: CB.

19. AB is any chord of a circle; AC, BC are drawn to any point C in the circumference and meet the diameter perpendicular to AB at D, E: if O be the centre, shew that the rect. OD, OE is equal to the square on the radius.

20. YD is a tangent to a circle drawn from a point Y in the diameter AB produced; from D a perpendicular DX is drawn to the diameter: shew that the points X, Y divide AB internally and externally in the same ratio.

21. Determine a point in the circumference of a circle, from which lines drawn to two other given points shall have a given ratio.

22. O is the centre and OA a radius of a given circle, and V is the middle point of OA; P and Q are two points on the circumference on opposite sides of A and equidistant from it; QV is produced to meet the circle in L: shew that, whatever be the length of the arc PQ, the chord LP will always meet OA produced in a fixed point.

23. EA, EA' are diameters of two circles touching each other externally at E; a chord AB of the former circle, when produced, touches the latter at C', while a chord A'B of the latter touches the former at C: prove that the rectangle, contained by AB and A'B', is four times as great as that contained by BC' and B'C.

24. If a circle be described touching externally two given circles, the straight line passing through the points of contact will intersect the line of centres of the given circles at a fixed point.

25. Two circles touch externally in C; if any point D be taken without them so that the radii AC, BC subtend equal angles at D, and DE, DF be tangents to the circles, shew that DC is a mean proportional between DE and DF.

26. If through the middle point of the base of a triangle any line be drawn intersecting one side of the triangle, the other produced, and the line drawn parallel to the base from the vertex, it will be divided harmonically.

27. If from either base angle of a triangle a line be drawn intersecting the median from the vertex, the opposite side, and the line drawn parallel to the base from the vertex, it will be divided harmonically.

28. Any straight line drawn to cut the arms of an angle and its internal and external bisectors is cut harmonically.

29. P, Q are harmonic conjugates of A and B, and C is an external point: if the angle PCQ is a right angle, shew that CP, CQ are the internal and external bisectors of the angle ACB.

30. From C, one of the base angles of a triangle, draw a straight line meeting AB in G, and a straight line through A parallel to the base in E, so that CE may be to EG in a given ratio.

313 P is a given point outside the angle formed by two given lines AB, AC: shew how to draw a straight line from P such that the parts of it intercepted between P and the lines AB, AC may have a given ratio.

32. Through a given point within a given circle, draw a straight line such that the parts of it intercepted between that point and the circumference may have a given ratio. How many solutions does the problem admit of?

33. If a common tangent be drawn to any number of circles which touch each other internally, and from any point of this tangent as a centre a circle be described, cutting the other circles; and if from this centre lines be drawn through the intersections of the circles, the segments of the lines within each circle shall be equal.

34.

APB is a quadrant of a circle, SPT a line touching it at P; C is the centre, and PM is perpendicular to CA: prove that

the A SCT: the A ACB :: the ▲ ACB: the A CMP.

35. ABC is a triangle inscribed in a circle, AD, AE are lines drawn to the base BC parallel to the tangents at B, C respectively; shew that AD AE, and BD: CE :: AB2: AC2.

36. AB is the diameter of a circle, E the middle point of the radius OB; on AE, EB as diameters circles are described; PQL is a common tangent meeting the circles at P and Q, and AB produced at L: shew that BL is equal to the radius of the smaller circle.