PROPOSITION 4. THEOREM. If in a circle two chords cut one another, which do not both pass through the centre, they cannot both be bisected at their point of intersection. A Let ABCD be a circle, and AC, BD two chords which intersect at E, but do not both pass through the centre. Then AC and BD shall not be both bisected at E. CASE I. If one chord passes through the centre, it is a diameter, and the centre is its middle point; .. it cannot be bisected by the other chord, which by hypothesis does not pass through the centre. CASE II. If neither chord passes through the centre; then, if possible, suppose E to be the middle point of both ; that is, let AE = EC; and BE = ED. Construction. Find F, the centre of the circle. Join EF. III. 1. Because FE, which passes through the centre, Proof. bisects the chord AC, ..the FEC is a rt. angle. Hyp. III. 3. And because FE, which passes through the centre, bi sects the chord BD, .. the ▲ FED is a rt. angle. .. the FEC = the FED, the whole equal to its part, which is impossible. .. AC and BD are not both bisected at E. Hyp. III. 3. Q.E.D. EXERCISES. ON PROPOSITION 1. 1. If two circles intersect at the points A, B, shew that the line which joins their centres bisects their common chord AB at right angles. 2. AB, AC are two equal chords of a circle; shew that the straight line which bisects the angle BAC passes through the centre. 3. Two chords of a circle are given in position and magnitude: find the centre of the circle. 4. Describe a circle that shall pass through three given points, which are not in the same straight line. 5. Find the locus of the centres of circles which pass through two given points. 6. Describe a circle that shall pass through two given points, and have a given radius. When is this impossible? ON PROPOSITION 2. 7. A straight line cannot cut a circle in more than two points. ON PROPOSITION 3. 8. Through a given point within a circle draw a chord which shall be bisected at that point. 9. The parts of a straight line intercepted between the circumferences of two concentric circles are equal. 10. The line joining the middle points of two parallel chords of a circle passes through the centre. 11. Find the locus of the middle points of a system of parallel chords drawn in a circle. 12. If two circles cut one another, any two parallel straight lines drawn through the points of intersection to cut the circles, are equal. 13. PQ and XY are two parallel chords in a circle: shew that the points of intersection of PX, QY, and of PY, QX, lie on the straight line which passes through the middle points of the given chords. PROPOSITION 5. THEOREM. If two circles cut one another, they cannot have the same centre. E B 8 Let the two O' AGC, BFC cut one another at C. Then they shall not have the same centre. Construction. If possible, let the two circles have the same centre; and let it be called E. Join EC; and from E draw any st. line to meet the ces at F and G. And because E is also the centre of the BFC, Hyp. .. EF EC. the whole equal to its part, which is impossible. Therefore the two circles have not the same centre. Q.E.D. EXERCISES. ON PROPOSITIONS 4 AND 5. 1. If a parallelogram can be inscribed in a circle, the point of intersection of its diagonals must be at the centre of the circle. 2. Rectangles are the only parallelograms that can be inscribed in a circle. 3. Two circles, which intersect at one point, must also intersect at another. PROPOSITION 6. THEOREM. If two circles touch one another internally, they cannot have the same centre. A E B at C. Let the two ○ ABC, DEC touch one another internally Then they shall not have the same centre. Construction. If possible, let the two circles have the same centre; and let it be called F Join FC; and from F draw any st. line to meet the ces at E and B. Proof. Because F is the centre of the ABC, Hyp. .. FB FC. And because F is the centre of the O DEC, Hyp. ... FB = FE, the whole equal to its part, which is impossible. Therefore the two circles have not the same centre. Q.E.D. NOTE. From Propositions 5 and 6 it is seen that circles, whose circumferences have any point in common, cannot be concentric, unless they coincide entirely. Conversely, the circumferences of concentric circles can have no point in common. PROPOSITION 7. THEOREM. If from any point within a circle which is not the centre, straight lines are drawn to the circumference, then the greatest is that which passes through the centre; and the least is the remaining part of the diameter. And of all other such lines, that which is nearer to the greatest is always greater than one more remote. And two equal straight lines, and only two, can be drawn from the given point to the circumference, one on each side of the diameter. K Let ABCD be a circle, and from F, any point within it which is not the centre, let FA, FB, FC, FG, and FD be drawn to the Oce, of which FA passes through E the centre, and FD is the remaining part of the diameter. Then of all these st. lines, (i) FA shall be the greatest; (ii) FD shall be the least; (iii) FB, which is nearer to FA, shall be greater than FC, which is more remote; (iv) also two, and only two, equal st. lines can be drawn from F to the Oce. Proof. (i) In the FEB, the two sides FE, EB are I. 20. together greater than the third side FB. But EB = EA, being radii of the circle; .. FE, EA are together greater than FB; that is, FA is greater than FB. |