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6. A fegment of a circle, is a figure contained under any arc and the chord of that arc.
7. A tangent to a circle, is a right line which paffes through a point in the circumference without cutting it.
8. Right lines, or chords, are faid to be equally diftant from the centre of a circle, when perpendiculars drawn to them from the centre are equal.
9. And the right line on which the greater perpendi◄ cular falls, is faid to be farther from the centre.
10. An angle in a segment, is that which is contained by two right lines, drawn from any point in the arc of the fegment, to the two extremities of the chord of that arc.
11. One circle is faid to touch another, when it paffes through a point in its circumference without cutting it.
PROP. I. PROBLEM.
To find the centre of a given circle,
Let ABC be the given circle; it is required to find its
Draw any chord AB, and bifect it in D (I. 10.); and through the point D draw CE át right angles to Ab (I, II.), and bifect it in F: then will the point F be the centre of the circle.
For if it be not, fome other point must be the centre, either in the line EC, or out of it.
But it cannot be any other point in the line EC, for if it were, two lines drawn from the centre of the circle to its circumference would be unequal, which is abfurd.
Neither can it be any point out of that line; for if it can, let G be that point; and join GA, GD and GB.
Then, because GA is equal to GB (I. Def. 13.), AD to DB (by Conft.), and GD common to each of the triangles AGD, BGD, the angle ADG will be equal to the angle BDG (I. 7.)
But when one line falls upon another, and makes the adjacent angles equal, thofe angles are, each of them, right angles (I. Def. 8 and 9.)
The angle ADG, therefore, is equal to the angle ADC (I. 8.), the whole to the part, which is abfurd; confequently no point but F can be the centre of the circle.
Q. E. D. COROLL. If any chord of a circle be bifected, a right line drawn through that point, perpendicular to the chord, will pass through the centre of the circle.
PROP. II. THEOREM.
If any two points be taken in the circumference of a circle, the chord, or right line which joins them, will fall wholly within the circle.
Let ABE be a circle, and A, B any two points in the circumference; then will the right line AB, which joins thofe points, fall wholly within the circle.
For find c, the centre of the circle ABE (III.1.), and join C, A, C, B; and through any point D, in AB, draw the right line CE, cutting the circumference in E.
Then, because CA is equal to CB (I. Def. 13.), the angle CAB will be equal to the angle CBA (I. 5.)
And, fince the outward angle CDB of the triangle ACD, is greater than the inward opposite angle CAB (I. 16.), it will also be greater than the angle CBA.
But the greater fide of every triangle is oppofite to the greater angle (I. 17.); whence CB, or its equal CE, will be greater than CD.
The point D, therefore, falls within the circle; and the fame may be fhewn of any other point in AB; confequently the whole line AB muft fall within the circle. Q. E. D.
PROP. III. THEOREM.
If a right line, which paffes through the centre of a circle, bifect a chord, it will be perpendicular to it; and if it be perpendicular to the chord, it will bifect it.
Let ABC be a circle, and CE a right line which passes through the centre D, and bifects the chord AB in E; then will CE be perpendicular to AB.
For join the points AD, DB:
Then, because AD is equal to DB (II. Def. 13.), ae to EB (by Hyp.), and ED common to each of the triangles ADE, BDE, the angle DEA will be equal to the angle DEB (I. 7.)
But one line is faid to be perpendicular to another, when it makes the angles on both fides of it equal to each other (I. Def.8.); confequently CE is perpendicular to the chord AB.
Again, let the right line DE be drawn from the centre D, perpendicular to the chord AB; then will AB be bifected in the point E.
For join the points AD, DB, as before:
Then, fince the angle DAB is equal to the angle DBA (I. 5.), and the angle AED to the angle DEB, (being each of them right angles) the angle ADE will also be equal to the angle EDB (I. 28. Cor. 1.)
And, because the triangles DEA, DEB are mutually equiangular, and have the fide DE common, the fide AE will also be equal to the fide EB (I. 21.); whence AB is bifected in the point E, as was to be shewn.
COROLL. If a right line be drawn from the vertex of an ifofceles triangle, to the middle of the base, it will be perpendicular to it; and if it be perpendicular to the base, it will bifect both it and the vertical angle.
PROP. IV. THEOREM.
If more than two equal right lines can be drawn from any point in a circle to the circumference, that point will be the centre.
Let ABDC be a circle, and o a point within it; then if any three right lines OA, OB, OC, drawn from the point o to the circumference, be equal to each other, that point will be the centre.