PROPERTIES OF CIRCLES. THEOREM XXXVI.—If any point D (fig. 56, p. II.), be taken in ACB, the diameter of a circle DB, that part of the diameter which passes through the centre, C, is greater than any other line, as DE, that can be drawn from D to the circumference of the circle; and D E is greater than any other line DF drawn from D to the circumference; but at a greater distance from D B.

For, join CE and C F, then as C E is equal to CB, DC and C E together are equal to DB; and, as the sum of DC and C E is greater than DE (Theorem XIII.), D B is greater than D E. Again as DC and CE are respectively equal to DC and CF, but the angle DCE is greater than the angle DCF, DE (Theorem XI.) is greater than DF.

THEOREM XXXVII.-An angle ACB (fig. 57, p. II.) at the centre of a circle is double the angle ADB at the circumference, when they both stand on the same arc A B.

For join D C, and produce it as to E; then (Theorem XIX.) the angle ACE is equal to the sum of the angles CAD, and CDA; but, as CA is equal to CD, the angles CAD and CDA are equal; the angle ACE is therefore double the angle ADC. For a like reason the angle BCE is double the angle BDC, consequently the angle AC B is double the angle A D B.

Cor. 1. As the angle AC B, at the centre, is measured by the arc A B on which it stands, the angle ADB at the circumference is measured by half the arc A B, on which it stands.

Cor. 2. All angles in the same segment of a circle, or standing on the same arc, are equal to each other.

Cor. 3. The angle in a semicircle is a right angle; for, when AC and C B become one straight line, the angle ACB becomes equal to two right angles, and the angle A D B, which is in that case an angle in a semicircle, is a right angle.

THEOREM XXXVIII.-The sum of any two opposite angles BAD, BCD, of a quadrilateral inscribed in a circle, is two right angles (fig. 58, p. II.)

For the three angles of the triangle A D B are together equal to two right angles (Theorem XIX. Cor. 1). But (Theorem XXXVII. Cor. 2), the angle A D B is equal to the angle AC B, and the angle ABD to the angle A CD, hence the angles BAD and BDC are together equal to two right angles.

Cor. If A B be produced, as to E, then (Theorem VI.) the angles D A B and DAE, together, make two right angles; hence the angle DAE is equal to the inward and opposite one BCD.

THEOREM XXXIX.-In any circle as ABCD (fig. 59, p. II.) parallel chords as A B, CD, intercept equal ones, AC and B D.

For, BC being joined, the angles ABC and BCD are equal (Theorem XV.), and hence (Theorem XXXVII. Cor. 1) half the arc A C is equal to half the arc BD, or the whole arc A C, to the arc BD.

Cor. If A B be conceived to move parallel to itself till it coincide with the tangent EG F, then the arcs CG and G D intercepted by a tangent and a chord parallel to it will appear to be equal.

THEOREM XL-If A B (fig. 60, p. II.) be a tangent to a circle, and AC a chord, drawn from the point of contact A, the angle DAC is equal to any angle in the alternate segment ADC.

For from C draw the chord DC parallel to AB, and join AD. Then, as the arcs AC and AD are equal (Theorem XXXIX. Cor.), the angles ACD and ADC are equal (Theorem XXXVII. Cor. 2). But (Theorem XV.) the angles ACD and BAC are equal; therefore the angles BAC and ACD are equal; and, as ADC is equal to any angle in the same segment, BAC is also equal to any angle in that segment.

THEOREM XLI.-The angle EAC (fig. 61, p. II.), formed by two lines cutting a circle, is measured by half the sum or half the difference of the intercepted arcs EC and BD, according as the point A is within or without the circle.

For join EB; then when A is within the circle, the angle EAC is the sum of the angles CBE and DEB; but when A is without the circle, the angle EAC is the difference of the angles C BE and D E B (Theorem XIX. Cors. 1 and 2). But (Theorem XXXI. Cor. 1) CBE is measured by half E C, and the angle D E B by half B D ; hence the truth of the proposition is manifest.

THEOREM XLII.-If A'B (fig. 62, p. II.), any chord of a circle, be bisected by a line C D drawn to the centre, CD will be perpendicular to AB; or if CD, drawn from the centre, be perpendicular to AB, A B will be bisected in D.

Draw the radii C A, C B, then A B is bisected in D, the triangles ACD and BCD will be identical (Theorem V.), and have the angles ADC and BDC, opposite to the equal sides A C and B C, equal to each other, and CD is therefore perpendicular to A B. Or again, if CD is perpendicular to A B, the angles CDA and CDB will be equal, and, because of the equal sides AC and CB, the angles CAD and CBD will be equal (Theorem III); hence the angles ACD and BCD will also be equal, and as CD is common to both triangles, the side A D will be equal to the side B D (Theorem II.), or A B is bisected in D.

Cor. The angles ACD and B C D are equal, therefore the arcs A E and B E are equal.

THEOREM XLIII.-If on A B, CD (fig. 63, p. II.), two chords in a circle, the perpendiculars EG and GF be drawn; then if the perpendiculars are equal, the chords are equal; and if the chords are equal, the perpendiculars are equal.

For join AG and GC, then (Theorem XLII.) the chords A B and C D are bisected in E and F, if therefore A B is equal to C D, A E is equal to CF. But (Theorem XXXI. Cor.) A G = AE,2+ EG2; and GC' (or A C2) C'F' + GF. Hence A E2 + EG2 = C F2 + GF2; and if from these equals the equal squares A E2 and CF be taken, we have EGG F, or EGGF. In the same way it may be shown that, when EG CF, AE = C F, and consequently A B = CD.

THEOREM XLIV.-A perpendicular as BC (fig. 64, p. II.), at the extremity B of the radius of a circle, is a tangent to the circle.

For from the centre A draw AC to any point C in the line BC; then as ABC is a right angle, ACB is less than a right angle; hence

(Theorem X. AC is greater than AB, or the point C is without the circle. In the same way it may be shown that BC meets the circle in the point B only, and it is therefore a tangent to the circle.

OF PROPORTION.

Proportion is the numerical relation which one quantity bears to another. Quantities between which proportion can exist must be of the same kind, as a line and a line, a surface and a surface, a solid and a solid, an angle and an angle.

A greater quantity is said to be a multiple of a less, when it contains the less a certain number of times without any remainder; and quantities so related are said to have the same relation to each other that unity has to the number which indicates how often the less is contained in the greater.

If a quantity, as A (fig. 66, p. II.), be contained exactly a certain number of times in another quantity B, the quantity A is said to measure the quantity B; and, if the same quantity A be contained exactly a certain number of times in another quantity C, A is also said to be a measure of C; and it is called a common measure of the quantities B and C. The quantities B and C will evidently bear the same relation to each other, that the numbers do which represent the multiple that each quantity is of the common measure A.

Again, if a quantity, D, be contained as often in another quantity, E, as A is contained in B, and as often in another quantity F, as A is contained in C, then the proportion that E has to F will be the same that B has to C, and the quantities B, C, E, and F, are said to be proportional quantities, a relation which is commonly expressed thus B: C:: E: F.

THEOREM XLV.-Any two quantities as A B, CD (fig. 67, p. II.), have the same proportion that their like multiples have.

Let A B be to CD as any number (say 3) to any other number (say 4); or let A B contain three such equal parts as those of which CD contains 4; and let Ef, fg, g F, be any like multiples A a, a b, and b B; and Gh, hi, ik, and k H, the same multiples of Cc, cd, de, and eD; then EF is the same multiple of A B that GH is of CD; and the same that each part of the one is of the corresponding part of the other. And, as the parts of AB and CD are equal, the like multiples of those parts which constitute the parts of EF and GH are also equal. Hence EF is to GH as 3 is to 4, the same proportion that A B has to CD. In the same way may the property be proved, whatever numerical relation AB may have to CD.

Cor. Quantities of the same kind are to each other as their like parts.

THEOREM XLVI.-In any four quantities AB, BC, DE, and EF (fig. 68, p. II.) of the same kind, if AB BC: DE: EF, then also alternately A B : DE:: BC: EF. If A B contain any number of such equal parts A a (say 4), as those of which BC contains any other number (say 3), Bb, then DE also will contain four such equal parts D d as those of which EF con

tains three, Ee. Then (Theorem XLV. Cor.)
AB will be to DE as Aa to Dd, and BC to
EF as Bb to Ee; or as Aa to Dd; hence
AB: DE:: BC: EF.

Cor. If AB: BC:: DE:EF; then BC:
AB::FE: ED.

THEOREM XLVII.-In any four quantities AB, BC, D E, and E F (fig. 68, p. II.), of the same kind, if AB: BC :: DE: EF; then AB: AB+ BC:: DE: DE+ EF.

For let A B BC, or DE: EF::m: n; then AB: AB BC, or DE: DE EF:m: mn. In the same way it may be shown that AB+BC: AB BC:: DE+EF: DE EF.

THEOREM XLVIII.-Triangles, as ABC, DEF (fig. 69, p. II.), between the same parallels AE, CF, or that have equal altitudes, are to each other as their bases, A B and D E.

For let A B be to DE as any number, (3, for example), to any other number, (as 4); that is, let AB contain three such equal parts, Aa, ab, b B, as those of which DE contains four, Dc, ce, ef, fE; and join Ca, Cb, Fc, Fe, and Ff.

Then the triangles CA a, Cab, CbB, F D c, Fce, &c., are all (Theorem XXV. Cor. 1) equal; therefore the triangle ABC contains three such equal parts as those of which the triangle D E F contains four. Hence the triangle ABC is to the triangle DEF as the base AB is to the base D E.

Cor. Parallelograms and rectangles between the same parallels, or that have equal altitudes, are to each other as their bases; for the parallelograms are doubles of their respective triangles.

THEOREM XLIX.-If two triangles, as A B C, DEF (fig. 70, p. II.), stand on equal bases, AB, DE, the triangles are to each other as their altitudes CH, FI.

Let BP be perpendicular to A B and equal to CH; in BP take BQ equal to F I, and join AP, AQ, and CP. Then the triangle AP B is equal to the triangle ABC, and the triangle ABQ to the triangle D F E. But ABP: ABQ:: BP: BQ (Theorem XLVIII.), therefore ABC: DEF::HC: FI.

Cor. Parallelograms and rectangles on equal bases are to each other as their altitudes.

THEOREM L.-If four lines, as A, B, C, D (fig. 71, p. II.), be proportional, the rectangle under the extremes A and D will be equal to the rectangle under the means B and C.

For AD: BD::A: B (Theorem XLIX. Cor.), and BC: B D:: C: D; consequently as A: B:: C: D, A D: BD:: B.C: BD; and therefore A D and B C are equal because they bear the same proportion to BD.

Cor. If the means are equal, the rectangle of the extremes will be equal to the square of the

THEOREM LI.—If of four lines, as A, B, C, D · (fig. 71, p. II.), the rectangle A D of two of them, be equal to the rectangle B C of the other two, then the sides of those rectangles will be inversely proportional; viz. A: B::C:D; or B: A:: D: C, or A: C:: B : D, &c.