101. Exercises

1. If one side of an inscribed quadrilateral be produced, the exterior thus formed at one vertex

equals the interior at the opposite vertex of the quadrilateral.

State and prove the converse.

2. From a point O without a circle two st. lines OAB, OCD are drawn cutting the circumference at A, B, C, D.

Show that As OBC, OAD are similar, and that As OAC, OBD are similar.

3. If a gm be inscribed in a circle, the gm is a rect. 4. A, D, C, E, B are five successive points on the circumference of a circle; and A, B are fixed. Show that the sum of the S ADC, CEB is the same for all positions of D, C, E.

5. A circle is circumscribed about an equilateral Show that the in each segment outside the ▲ is an of 120°.

Δ.

6. A scalene ▲ is inscribed in a circle. Show that the sum of the s in the three segments outside the ▲ is 360°.

7. A quadrilateral is inscribed in a circle. Show that the sum of the s in the four segments outside the quadrilateral is 540°.

8. P is a point on the diagonal KM of the gm KLMN. Circles are described about PKN and PLM. Show that LN passes through the other point of intersection of the circles. 9. A circle drawn through the middle points of the sides of a passes through the feet of the Ls from the vertices to the opposite sides.

10. If the opposite sides of a quadrilateral inscribed in & circle be produced to meet at L and M, and about the ▲s

so formed outside the quadrilateral circles be described intersecting again at N, then L, M, N are in the same st. line.

11. In a ▲ DEF, DX LEF and EY DF. Prove that L XYF = L DEF.

12. PQRS, PQTV are circles and SPV, RQT are st. lines. Prove that SR || VT.

13. The st. lines that bisect any of a quadrilateral inscribed in a circle and the opposite exterior meet on the circumference.

14. XYZ is a ▲ ; YD ¦ ZX, and DE 1 XY; ZF 1 XY and FG ZX. Show that EG || YZ.

Show

15. EGD, FGD are two circles with centres H, K respectively. EGF is a st. line. EH, FK meet at P. that H, K, D, P are concyclic.

16. KL, MN are two || chords in a circle; KE, NF two chords in the same circle. Show that LF ME.

17. The bisectors of the s formed by producing the opposite sides of a quadrilateral inscribed in a circle are to each other.

18. HKM, LKM are two circles, and HKL is a st. line. HM, LM cut the circles again at E, F respectively, and HF cuts LE at G. Show that a circle may be circumscribed about MEGF.

19. PQRS is a quadrilateral and the bisectors of the ▲s P, Q; Q, R; R, S; S, P meet at four points. Show that a circle may be circumscribed about the quadrilateral thus formed.

20. EF is the diameter of a semi-circle and G, H any two points on its arc. EH, FG cut at K and EG, FH cut at L. Show that KL EF.

21. DE is the diameter, O the centre and P any point on the arc of a semi-circle. PMLDE. Show that the bisector of MPO passes through a fixed point.

22. PQR is a ▲ and PDQ, PFQ are two circles cutting PR at D, F and QR at E, G. Prove that DE || FG.

THEOREM 13

If two angles at the centre of a circle are equal to each other, they are subtended by equal arcs.

Hypothesis. AKC, DKF are equal Ls at the centre

K of the circle ACD.

To prove that arc AEC equals arc DGF.

Construction.

L CKD.

Draw the diameter HKL bisecting

Proof-Suppose the circle to be folded along the diameter HKL, and the semi-circle HFL will coincide throughout with the semi-circle HAL.

.. the arc DGF coincides with the arc CEA.

.. arc DGF: = arc CEA.

102. Exercises

1. If two arcs of a circle be equal to each other, they subtend equals at the centre. (Prove either by indirect demonstration, or by the construction and method used in III-13.)

2. If two Ls at the circumference of a circle be equal to each other, they are subtended by equal arcs.

3. If two arcs of a circle be equal to each other, they subtend equal ▲s at the circumference.

4. In equal circles equals at the centres (or circumferences) stand on equal arcs.

5. In equal circles equal arcs subtend equals at the centres (or circumferences).

6. If two arcs of a circle (or of equal circles) be equal, they are cut off by equal chords.

7. If two chords of a circle be equal to each other, the major and minor arcs cut off by one are respectively equal to the major and minor arcs cut off by the other.

8. If two sectors of a circle have equals at the centre, the sectors are congruent.

9. Bisect a given arc of a circle.

10. Parallel chords of a circle intercept equal arcs. Show also that the converse is true.

11. If two equal circles cut one another, any st. line drawn through one of the points of intersection will meet the circles again at two points which are equally distant from the other point of intersection.

12. The bisectors of the opposites of a quadrilateral inscribed in a circle meet the circumference at the ends of a diameter.

13. If two s at the centre of a circle be supplementary, the sum of the arcs on which they stand is equal to half the circumference.

14. If any number of s be in a segment, their bisec tors all pass through one point.

TANGENTS AND CHORDS

103. Definitions. Any straight line which cuts a circle is called a secant.

A straight line which, however far it may be produced, has one point on the circumference of a circle, and all other points without the circle is called a tangent to the circle.

A tangent is said to touch the circle.

The common point of a tangent and circle, that is, the point where the tangent touches the circle, is called the point of contact.

B

E

ABC is a secant drawn to the circle BCF from the point A.

DFE is a tangent to the circle BCF, touching the circle at the point of contact F.

If the secant ABC rotate about the point A until the two points B, C where it cuts the circle coincide at G, the secant becomes a tangent having G for the point of contact.