PROPOSITION 36. THEOREM.

If from a 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 shall be equal to the square of the line which touches it.

From any pt. D without the ADC, let a tangent DB and a secant DCA be drawn; then rect. AD, DC=sq. on DB.

(1) If DCA passes through the centre E, join EB.

B

A

Rect. AD, DC with sq. on EC=sq. on ED,

[II. 6.

=sqs. on EB, BD.

[I. 47.

But sq. on EC=sq. on EB (.. EC=EB),

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

(2) If DCA does not pass through the centre E, join EB, EC,

and draw EF r to AC.

Then AFFC,

... rect. AD, DC with sq. on FC=sq. on FD,

[III. 3. [II. 6.

... rect. AD, DC with sqs. on FC, FE=sqs. on FD, FE,

=sqs. on EB, BD.

[I. 47. [I. 47.

... rect. AD, DC with sq. on EC=sq. on ED,

B

E

But sq. on EC=sq. on EB (·.· EC=EB), ... rect. AD, DC=sq. on BD.

COROLLARY.—(i.) If from any point without a circle there be drawn two straight lines cutting it, the rectangles contained by the whole lines and the parts of them without the circle are equal to one another.

For each is equal to the square of the tangent to the circle from the same point. (See also Ex. 195.)

COR.-(ii.) If two tangents to a circle be drawn from the same point, they are equal.

Ex. 374.-(i.) If the common chord of two intersecting circles is produced to any point, the tangents to the two circles from this point are equal.

(ii.) If the tangents to two intersecting circles from any point be equal, that point must be on the common chord produced.

Hence :-The locus of a point from which equal tangents can be drawn to two given intersecting circles consists of the parts external to the circles of the line passing through the points of section.

For an important extension to this theorem, see page 240 (Radical Axis). Ex. 375.-If the common chord of two intersecting circles be produced to cut a common tangent, it will bisect it.

Ex. 376.-The three common chords of three circles which intersect each other are concurrent.

S

Ex. 377.-Deduce III. 36 from III. 35.

Produce BD, CD to B', C', so that DB'= DB and DC'= CD, and show that A, B, C', B' are concyclic.

Ex. 378.-Deduce III. 36, Cor. (i.), from III. 35.

Ex. 379.-In equiangular triangles, the rectangles contained by the non-corresponding sides about equal angles are equal.

Let ABC, AB'C' be equiangular triangles, and let them be placed so that AB' falls along AC, and therefore AC' along AB. Show that B, C, B' C' are concyclic, and then use III. 36, Cor. (i.).

Prove the same theorem also by III. 35.

Ex. 380.-Demonstrate the converse of III. 36, Cor. i.—(i.) indirectly; (ii.) by the converse of III. 35.

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 straight line which cuts the circle and the part of it without the circle be equal to the square of the line which meets the circle, the line which meets the circle shall touch it.

From a pt. D without the ABC, let there be two st. lines, DB, DCA, drawn to the circle, and let rect. AD, DC= sq. on DB, then DB shall touch the circle

Draw the tangent DE.

Find the centre F and join FE, FB, FD.
Then sq. on DB=rect. AD, DC,

... FBD is a rt. 4,

... DB is the tangent at B.

Ex. 381.-Prove III. 37 indirectly.

[HYP. [III. 36.

Ex. 382.—CR is a common chord of two circles; P any point on CR

produced; MN any chord of either circle which produced; PA is a tangent to the other circle. circle of triangle AMN touches the latter.

passes through P when Prove that the circum

DEFINITIONS.

Book III.

I. Equal circles are those of which the diameters are equal, or from the centres of which the straight lines to the circumferences are equal.

'This is not a definition, but a theorem, the truth of which is evident; for, if the circles be applied to one another, so that their centres coincide, the circles must likewise coincide, since the straight lines from the centres are equal.'

II. A straight line is said to 'touch' a circle, when it meets the circle, and being produced does not cut it.

III. Circles are said to 'touch' one another, which meet, but do not cut one another.

IV. Straight lines are said to be equally distant from the centre' of a circle, when the perpendiculars drawn to them from the centre are equal.

V. And the straight line on which the greater perpendicular falls is said to be 'farther from the centre.'

VI. A 'segment of a circle' is the figure contained by a straight line and the circumference it cuts off.

VII. 'The "angle of a segment" is that which is contained by the straight line and the circumference.'

VIII. An 'angle in a segment' is the angle contained by two straight lines drawn from any point in the circumference of the segment, to the extremities of the straight line which is the base of the segment.

IX. And an angle is said to insist or 'stand upon' the circumference intercepted between the straight lines that contain the angle.

X. A 'sector' of a circle is the figure contained by two straight lines drawn from the centre, and the circumference between them.

XI. 'Similar segments' of circles are those in which the angles are equal, or which contain equal angles.