A treatise on navigation, and nautical astronomy

BCE is double of BDC; and consequently the angle A C B is double the angle A DB. Q. E. D.

Cor. 1. As the angle AC B, at the centre, is measured by the arc A B on which it stands, the angle A D B, at the circumference, is measured by half the arc AB 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; for each of them is equal to half the angle at the centre, standing on the same arc.

Cor. 3. The angle in a semicircle is a right angle. For when AC and CB (first fig. of this prop.) become one straight line, A D B will be a semicircle, and the angles ACE and BCE together, will be equal to two right angles; whence A D B, which is half the sum of ACE and BCE, will in that case be equal to one right angle.

THEOREM LIV.

The sum of any two opposite angles, B AC, and BCD, of a quadrangle ABCD inscribed in a circle, is equal to two right angles.

For the angle B A C is measured by half the arc BCD, and the angle B CD by half the arc BAD; (Cor. 1. Theo. 53.) therefore the sum of the angles BAC and BCD, is measured by half the whole circumference. But half the whole circumference is the measure of two right angles. Hence the angles B A C and BCD are, together, equal to two right angles. Q. E. D.

THEOREM LV.

E A

If any side B A, of a quadrilateral, ABCD, (see the last figure) inscribed in a circle, be produced, the outward angle, DAE, will be equal to the inward and opposite angle, C.

For the angles DAB and DAE are equal to two right angles, (Theo. 6.) and the angles D A B and D C B are also equal to two right angles. (Theo. 54.) Hence by omitting the common angle DAB from each sum, the remaining angles, DAE and DC B, are equal. Q. E.D.

THEOREM LVI.

If, in a circle A B D C, there be drawn two parallel chords A B, CD; the intercepted arcs AC and B D will be equal.

For join BC. Then because AB and CD are parallel, the alternate angles A B C and DCB are equal. (Theo. 17.) But the angle ABC is measured by half the arc A C, and the angle DC B, by half the arc D B. (Cor. 1. Theo. 53.) Whence the halves of AC and BD are equal, and consequently the whole arcs A C and BD are equal. Q. E. D.

Cor. If A B move parallel to itself, till it coincide, at G, with the tangent EGF, the intercepted arcs, G C and G D, will be equal.

THEOREM LVII.

If AB be a tangent to a circle, and AC a chord drawn from A, the point of contact, the angle BAC will be equal to any angle in the alternate segment ADC.

For from C draw the chord D C parallel to the tangent A B, and join A D. Then as the arcs A C and AD are equal, (Theo. 56.) the angles ADC and A C D are equal, being measured by the halves of the equal arcs AC and A D. (Cor. 1. Theo. 53.) But the angle A CD is equal to the alternate angle BAC (Theo. 17.); therefore the angle B A C is equal to the angle AD C. And as the angle A D C is equal to any angle in the segment ADC (Theo. 53. Cor. 2.), the angle B A C is also equal to any angle in the alternate segment A DC. Q. E.D.

Cor. As the angle ADC is measured by half the arc A C, the angle B AC, made by the tangent A B, and the chord A C, is also measured by half the intercepted arc A C.

THEOREM LVIII.

The angle DE B formed within any circle A B C D, by the intersection of two chords AB, CD, is measured by half the sum of the intercepted arcs A C and BD.

For join A D. Then the angle ADE or AD C is measured by half the arc A C, and the angle D A E, or DAB, by half the arc B D. (Cor. 1. Theo. 53.) But the angle DEB is equal to the sum of the angles DAE and ADE (Theo. 22.) and it is therefore measured by half the sum of the arcs AC and BD.

;

Q. E. D.

THEOREM LIX.

If two chords A B, CD, of the circle A B C D, meet when produced in a point E without the circle, the angle E is measured by half the difference of the intercepted arcs AC and B D.

For join A D. Then the angle ADC is measured by half the arc A C, and the angle D A B by half the arc D B. (Cor. 1. Theo. 53.) But the angle E is equal to the difference of the angles DAB and EDA (Theo. 22. Cor.); and therefore the angle D E B is measured by half the difference of the arcs AC and BD. Q.E.D.

Cor. If one of the lines, as E C D, revolve round the point E, till the points C and D coincide in F, EF will then be a tangent to the circle, and the angle FEA will be measured by half the difference of the intercepted arcs A F and F B.

THEOREM LX.

If AB, any chord of a circle, be bisected in D, and the point D be joined to C, the centre of the circle; then C D will be perpendicular to AB; or if CD, drawn from the centre, be perpendicular to the chord A B, A B will be bisected in D.

For let the two radii CA, CB be drawn, then if AB be bisected in D, the two triangles A CD, BCD, will have the sides A C and B C equal, AD and BD equal, and D C common to both; they are therefore identical (Theo. 5.), and have the adjacent angles ADC and BDC equal; these angles are consequently right angles, and CD is therefore perpendicular to A B. Again if C D be perpendicular to A B, the angles C D A and C D B will be equal; and C A being equal to C B, the angle C A B, or CAD is equal to the angle C BA, or CBD (Theo. 3.); therefore A CD and BCD, the remaining angles of each triangle, will also be equal (Theo. 24. Cor. 1.) Hence, as AC is equal to C B, the triangles AD C and BDC will be identical (Theo. 2.), and consequently the side AD will be equal to the corresponding side B D; or A B is bisected in D. Q. E. D.

Since the angle ACD is equal to the angle BCD, the arc AE will be equal to the arc B E.

THEOREM LXI.

Let A B, C D be any two chords in a circle AB DC, and let GE, GF be perpendiculars drawn from the centre G, on the chords A B, and C D. If A B and C D are equal, E G and G F are equal; or if E G and G F are equal, A B and C D are also equal.

and B

А_С

For draw the two radii A G, GC; then as A B and CD are bisected in E and F (Theo. 60.), if they are equal, their halves A E and C B will be equal. But the square of AG is equal to the sum of the squares of A E and E G, and the square of G C is equal to the sum of the squares of G F and FG (Theo. 44.); as A G and G C are equal, their squares are equal. of A E and E G are together equal to the sum of the squares of C F and F G ; and if the squares of the equal lines A E and C F be taken from each sum, the square of E G will remain equal to the square of GF, and consequently GE will be equal to G F. In the same way

Hence the

squares

it may be shewn, that when G E is equal to GF, AE is equal to CF, therefore A B and C D, the doubles of those lines, are equal.

THEOREM LXII.

If BD and B C be two unequal chords in a circle, the angle B AD at the centre, subtended by the greater chord BD is greater than the angle BAC subtended by the less.

For as the sides AD, A B, of the triangle D A B are equal to the sides AC, AB, of the triangle CAB, but DB is greater than B C, therefore the angle DAB is greater than the angle CAB. D (Theo. 14.)

Cor. 1. As the angle D A B is measured by the arc DC B, and the angle CAB by the arc BC, therefore the arc DC B, subtended by the greater chord, is greater than the arc B C, subtended by the less. Cor. 2. In the same circle equal chords subtend equal arcs, or equal angles, whether at the centre or circumference; and equal arcs, or equal angles, whether at the centre or the circumference, are subtended by equal chords.

THEOREM LXIII.

If A CB, an angle at the centre of a circle, be the sixth part of four right angles, or the third part of two right angles, AB, the chord of the arc which measures the angle AC B, will be equal to the radius of the circle.

For as all the interior angles of a triangle are, together, equal to two right angles, (Theo. 24.) if A CB be the third part of two right angles, the remaining angles A and B will, together, be equal to two-thirds of two right angles. And as AC and B C are equal, the angles A and B are equal (Theo. 3.); and consequently each of these is equal to the third part of two right angles, or to the angle ACB. Hence the triangle ABC being equiangular, is equilateral, (Cor. Theo. 4.) or A B is equal to AC or B C.

Remark. The circumference of a circle, or the measure of four right angles, is commonly divided, in practice, into 360 equal parts, called degrees; therefore the chord of 60 degrees, or the chord of the measure of the sixth part of four right angles, is in any circle, equal to the radius; and hence if the chords of every arch in a circle of a given radius, were arranged on a line, such a line, called a line of chords, would afford a convenient practical method of describing an angle, whose measure might be given in degrees, &c.

THEOREM LXIV.

If AB be the radius of a circle, then BC, a perpendicular on its extremity B, will be a tangent to the circle.

For from any other point C, in the line B C, draw CDA to the centre A, meeting the circle in D. Then because the angle ABC is a right angle, the angle ACB is less than a right angle (Theo. 24.); therefore AC is greater than A B (Theo. 12.), or than its equal AD; and consequently the point C is without the circle. In the same way every other point of the line BC, except the point B, may be shewn to be without the circle; and therefore the line BC meets the circle in the point B only, and it is consequently a tangent to the circle. Q. E. D.

THEOREM LXV.

If BC (see the last figure) touch the circumference of the circle in B, the radius AB will be perpendicular to BC.

For as BC touches the circumference in the point B only, every other point of BC will be without the circle; A B is therefore the shortest line that can be drawn from the point A to the line BC; hence A B is perpendicular to B C. (Theo. 27.)

THEOREM LXVI.

If BD touch the circumference of a circle in B, then BA drawn perpendicular to B C, will pass through the centre of the circle.

A C

For if the centre of the circle be not in the line AB, let any point C out of that line be the centre of the circle, and join BC. Then the angle CBD is a right angle (Theo. 64.), and it is consequently equal to ABD, which is also, by condition, a right angle; that is, the less, angle is equal to the greater, which is impossible. Hence, no point out of the line BA can be the centre of the circle, and the line BA therefore passes through the centre. Q. E. D.

THEOREM LXVII.

B D

In any circle ACDB, if the chord C D, and the diameter A B, meet each other in G, the rectangle of G A, G B, the segments of the one, will be equal to the rectangle GC, G D, the segments of the other.

For let DE be joined, and the perpendicular E F drawn to D C. Then because DE is equal to E B, GB is the sum of G E and ED; and because D E is equal to E A, A G is the difference of D E and E G. Again, because D F is equal to FC D (Theo. 60.), G C is the difference of

the segments of the base D F and

segments D F and F G.

FG, made by the perpendicular EF; and D G is the sum of the

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