PROPOSITION XXXIV. PROBLEM.

To cut off a segment from a given circle, capable of containing an angle equal to a given angle.

Let ABC be the given O, and D the given

It is required to cut off from ABC a segment capable of containing an L = L D.

Draw the st. line EBF to touch the circle at B.

Then the chord BC is drawn from the pt. of contact B,

.. FBC = 4 in segment BAC,

III. 32.

that is, the segment BAC contains an =D; and .. a segment has been cut off from the O, as was reqd.

Q. E. F

Ex. 1. If two circles touch internally at a point, any straight line passing through the point will divide the circles into segments, capable of containing equal angles.

Ex. 2. Given a side of a triangle, its vertical angle, and the radius of the circumscribing circle: construct the triangle.

Ex. 3. Given the base, vertical angle, and the perpendicular from the extremity of the base on the opposite side: construct the triangle.

PROPOSITION XXXV. THEOREM.

If two chords in a circle cul one another, the rectangle contained by the segments of one of them, is equal to the rectangle contained by the segments of the other.

Let the chords AC, BD in the ABCD intersect in the pt. P. Then must rect. AP, PC-rect. BP, PD.

From O, the centre, draw OM, ON 1s to AC, BD,

Then

and join OA, OB, OP.

II. 5.

AC is divided equally in M and unequally in P, .. rect. AP, PC with sq. on MP=sq. on AM. Adding to each the sq. on MO,

rect. AP, PC with sqq. on MP, MO=sqq. on AM, MO; .. rect. AP, PC with sq. on OP=sq. on OA.

In the same way it may be shewn that

rect. BP, PD with sq. on OP=sq. on OB.
Then sq. on OA=sq. on OB,

I. 47.

..rect. AP, PC with sq. on OP=rect. BP, PD with sq. on OP;

.'. rect. AP. PC=rect. BP, PD.

Q. E. D.

Ex. 1. A and B are fixed points, and two circles are described passing through them; PCQ, P'CQ' are chords of these circles intersecting in C, a point in AB; shew that the rectangle CP, CQ is equal to the rectangle CP', CQ'.

Ex. 2. If through any point in the common chord of two circles, which intersect one another, there be drawn any two other chords, one in each circle, their four extremities shall all lie in the circumference of a circle.

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.

D

M

Let D be any pt. without the ABC,

and let the st. lines DBA, DC be drawn to cut and touch the . Then must rect. AD, DB=sq. on DC.

From O, the centre, draw OM bisecting AB in M,

Then

and join OB, OC, OD.

AB is bisected in M and produced to D,

.. rect. AD, DB with sq. on MB=sq. on MD. II. 6. Adding to each the sq. on MO,

rect. AD, DB with sqq. on MB, MO=sqq. on MD, MO. Now the angles at M and C are rt. ≤s; III. 3 and 18. .. rect. AD, DB with sq. on OB=sq. on OD;

... rect. AD, DB with sq. on OB=sqq. on OC, DC. I. 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 on the line which meets it, the line which meets must touch the circle. A

B

D

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.
ACD cuts the O, and AE touches it,
.. rect. DA, AC=sq. on AE.

Then

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.

But 4 AEO is a rt. 4;

L

..LABO is a rt.

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

.. AB touches the O.

III. 36.

Hyp.

I. c.

III. 18.

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 F and 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 be 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 distances of