Ex. 1. If a circle be described on the radius of another circle as diameter, any straight line, drawn from the point, where they meet, to the outer circumference, is bisected by the interior one.

Ex. 2. If a straight line be drawn to touch a circle, and be parallel to a chord, the point of contact will be the middle point of the arc cut off by the chord.

Ex. 3. If, from any point without a circle, lines be drawn touching it, the angles contained by the tangents is double of the angle contained by the line joining the points of contact, and the diameter drawn through one of them.

Ex. 4. The vertical angle of any oblique-angled triangle inscribed in a circle is greater or less than a right angle, by the angle contained by the base and the diameter drawn from the extremity of the base.

Ex. 5. If, from the extremities of any diameter of a given circle, perpendiculars be drawn to any chord of the circle that is not parallel to the diameter, the less perpendicular shall be equal to that segment of the greater, which is contained between the circumference and the chord.

Ex. 6. If two circles cut one another, and from either point of intersection diameters be drawn, the extremities of these diameters and the other point of intersection lie in the same straight line.

Ex. 7. Draw a straight line cutting two concentric circles, so that the part of it which is intercepted by the circumference of the greater may be twice the part intercepted by the circumference of the less.

Ex. 8. Describe a square equal to the difference of two given squares.

Ex. 9. 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 all be parallel.

PROPOSITION XXXII. THEOREM.

If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles made by this line with the line touching the circle must be equal to the angles, which are in the alternate segments of the circle.

Let the st. line AB touch the O CDEF in F.
Draw the chord FD, dividing the into segments FCD,FED.
Then must DFB = 4 in segment FCD,

Take any pt. E in the arc FED, and join FE, ED, DC.

Then

III. 19.

FDC is a semicircle, .. FDC is a rt. 4;

III. 31.

.. sum of 48 FCD, CFD=a rt. 4.
Also, sum of 48 DFB, CFD=art. 4;

.. sum of 28 DFB, CFD= sum of 48 FCD, CFD,
and .. DFB= 1 FCD,

I. 32.

Again, CDEF is a quadrilateral fig. inscribed in a O, .. sum of 48 FED, FCD=two rt. 4 s.

Also, sum of 48 DFA, DFB=two rt. 48;

III. 22.

I. 13.

.: sum of 48 DFA, DFB=sum of 18 FED, FCD;
DFB has been proved = ▲ FCD ;

and

:. LDFA= L FED,

that is, ¿DFA= ▲ in segment FED.

Q. E.D.

Ex. The chord joining the points of contact of parallel

tangents is a diameter.

PROPOSITION XXXIII.

PROBLEM.

On a given straight line to describe a segment of a circle containing an angle equal to a given angle.

Let AB be the given st. line, and C the given 4. It is reqd. to describe on AB a segment of a contain an = 4 C.

which shall

At pt. A in st. line AB make ▲ BAD= ▲ C.
Draw AEL to AD, and bisect AB in F.
From F draw FG 1 to AB, meeting AE in G.
Then in As AGF; BGF;

.. AF=BF, and FG is common, and ▲ AFG= ▲ BFG; .. GA=GB.

I. 4.

With G as centre and GA as radius describe a O ABH.
Then will AHB be the segment reqd.

For AD is to AE, a line passing through the centre,
.. AD is a tangent to the © ABH.

III. 16. And the chord AB is drawn from the pt. of contact A, :. ¿BAD= ▲ in segment AHB,

that is, the segment AHB contains an 4 = 4
LC,
and it is described on AB, as was reqd.

III. 32.

Q. E. F.

Ex. 1. Two circles intersect in A, and through A is drawn a straight line meeting the circles again in P, Q. Prove that the angle between the tangents at P and Q is equal to the angle between the tangents at A.

Ex. 2. From two given points on the same side of a straight line, given in position, draw two straight lines which shall contain a given angle, and be terminated in the given line.

PROPOSITION XXXIV. PROBLEM.

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

B

Let ABC be the given O, and D the given 2.

It is reqd. to cut off from ABC a segment capable of containing an L = 4 D.

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

At B make L FBC= LD.

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

.. LFBC= 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 cut one another, the rectangle contained by the segments of one of them, is equal to the rectangle contained by the segments of the other.

M

P

Let the chords AC, BD in the O ABCD, intersect in the pt. P..

Then must rect. AP, PC=rect. BP, PD.

From O, the centre, draw OM, ON 18 to AC, BD,
and join OA, OB, OP.

Then 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,

IL. 5.

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,

L. 47.

.:. rect. AP, PC with sq. on OP=rect. BP, PD with 8q. 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, PCQ 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.