PROPOSITION XXXVI. THEOREM.

If, from any point without a circle, two straight lines be drawn, one of which cuts the circle, and the other touches it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, must be equal to the square on the line which touches it.

and let the st. lines DBA, DC be drawn to cut and touch the O.

Then

Then must rect. AD, DB=sq. on DC.

From O, the centre, draw OM 1 to AB,
and join OB, OC, OD.

AB is bisected in M and produced to D,
.. rect. AD, DB with sq. on MB=sq. on MD.
Adding to each the sq. on MO,

II. 6.

rect. AD, DB with sqq. on MB, MO=sqq. on MD, MO.
Now the angles at M and Care rt. 48; III. 3 and 17.
.. rect. AD, DB with sq. on OB=sq. on OD";
.. rect. AD, DB with sq. on OB=sqq. on OC, DC. L. 47.
And sq. on OB=sq. on OC;

.. rect. AD, DB=sq. on DC.

Q. E. D.

Ex. 1. Two circles intersect in A and B; shew that AB produced bisects their common tangent.

Ex. 2. If the circle, inscribed in a triangle ABC, touch BC in D, the circles described about ABD, ACD will touch each other.

PROPOSITION XXXVII. THEOREM.

If, from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it; if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square of the line which meets it, the line which meets must touch the circle.

Let A be a pt. without the

BCD, of which O is the centre. From A let two st. lines ACD, AB be drawn, of which ACD cuts the O and AB meets it.

Then if rect. DA, AC=sq. on AB, AB must touch the . Draw AE touching the C in E, and join OB, OA, OE.

Then : ACD cuts the ©, and AE touches it,

.. rect. DA, AC=sq. on AE.

But rect. DA, AC=sq. on AB;
.. sq. on AB=sq. on AE;
.. AB=AE.

Then in the As OAB, OAE,

* OB=OE, and OA is common, and AB=AE,

.LABO= = LAEO.

III. 36.
Нур.

L. C.

Now BO, if produced, is a diameter of the ";

.. AB touches the O.

III. 16.

Q. E.D.

Ex. If two circles cut each other, and from any point in the straight line produced, which joins their intersections, two tangents be drawn, one to each circle, they shall be equal.

Miscellaneous Exercises on Book III,

1. The segments into which a circle is cut by any straight line, contain angles whose difference is equal to the inclination to each other of the straight lines touching the circle at the extremities of the straight line which divides the circle.

2. If from the point in which a number of circles touch each other, a straight line be drawn cutting all the circles, shew that the lines which join the points of intersection in each circle with its centre will be all parallel.

3. From a point Q in a circle, QN is drawn perpendicular to a chord PP', and QM perpendicular to the tangent at P: shew that the triangles NQP', QPM are equiangular,

4. If a circle be described round the triangle ABC, and a straight line be drawn bisecting the angle BAC and cutting the circle in D, shew that the angle DCB will equal half the angle BAC.

5. One angle of a quadrilateral figure inscribed in a circle is a right angle, and from the centre of the circle perpendiculars are drawn to the sides, shew that the sum of their squares is equal to twice the square of the radius.

6. AB is the diameter of a semicircle, D and E any two points on its circumference. Shew that if the chords joining A and B with D and E, either way, intersect in Fand G, the tangents at D and E meet in the middle point of the line FG, and that FG produced is at right angles to AB.

7. If a straight line in a circle not passing through the centre be bisected by another and this by a third and so on, prove that the points of bisection continually approach the centre of the circle.

8. If a circle be described passing through the opposite angles of a parallelogram, and cutting the four sides, and the points of intersection joined so as to form a hexagon, the straight lines thus drawn shall be parallel to each other.

9. If two circles touch each other externally and any third circle touch both, prove that the difference of the dis

S. E. II.

12

tances of the centre of the third circle from the centres of the other two is invariable.

10. Draw two concentric circles, such that those chords of the outer circle, which touch the inner, may equal its diameter.

11. If the sides of a quadrilateral inscribed in a circle be bisected and the middle points of adjacent sides joined, the circles described about the triangles thus formed are all equal and all touch the original circle.

12. Draw a tangent to a circle which shall be parallel to a given finite straight line.

13. Describe a circle, which shall have a given radius, and its centre in a given straight line, and shall also touch another straight line, inclined at a given angle to the former.

14. Find a point in the diameter produced of a given circle, from which, if a tangent be drawn to the circle, it shall be equal to a given straight line.

15. Two equal circles intersect in the points A, B, and through B a straight line CBM is drawn cutting them again in C, M. Shew that if with centre C and radius BM a circle be described, it will cut the circle ABC in a point L such that arc AL arc AB.

=

Shew also that LB is the tangent at B.

16. AB is any chord and AC a tangent to a circle at A; CDE a line cutting the circle in D and E and parallel to AB. Shew that the triangle ACD is equiangular to the triangle EAB.

17. Two equal circles cut one another in the points A, B ; BC is a chord equal to AB; shew that AC is a tangent to the other circle.

18. In any two circles, which cut one another, the straight line joining the extremities of any two parallel radii cuts the line joining the centres in the same point.

19. A, B are two points; with centre B describe a circle, such that its tangent from A shall be equal to a given line.

20. If perpendiculars be dropped from the angular points of a triangle on the opposite sides, shew that the sum of the squares of the sides of the triangle is equal to twice the sum of the rectangles contained by the perpendiculars and that part of each intercepted between the angles of the triangles and the point of intersection of the perpendiculars.

21. When two circles intersect, their common chord bisects their common tangent.

22. Two circles intersect in A and B. Two points Cand D are taken on one of the circles; CA, CB meet the other circle in E, F, and DA, DB meet it in G, H: shew that FG is parallel to EH, and FH to EG.

23. A and B are fixed points, and two circles are described passing through them; CP, CP' are drawn from a point Con AB produced, to touch the circles in P, P'; shew that CP=CP.

24. From each angular point of a triangle a perpendicular is let fall upon the opposite side; prove that the rectangles, contained by the segments into which each perpendicular is divided by the point of intersection of the three, are equal to each other.

25. If from a point without a circle two equal straight lines be drawn to the circumference and produced, shew that they will be at the same distance from the centre.

26. Let O, O' be the centres of two circles which cut each other in A, A'. Let B, B' be two points, taken one on each circumference. Let C, C' be the centres of the circles BAB, BA'B'. Then prove that the angle CBC′ is equal to the angle Ο Α' Ο'.

27. The common chord of two circles is produced to any point P; PA touches one of the circles in A; PBC is any chord of the other: shew that the circle which passes through A, B, C touches the circle to which PA is a tangent.

28. Given the base of a triangle, the vertical angle, and the length of the line drawn from the vertex to the middle point of the base: construct the triangle.

29. If a circle be described about the triangle ABC, and a straight line be drawn bisecting the angle BAC and cutting