1. In a circle with centre O, two chords AC and BD intersect
at P. Show that
∠APB = ½ (∠AOB + ∠COD).

2. If the points A, B, C and D are any 4 points on a circle and P, Q, R and S are the midpoints of the arcs AB, BC, CD and
DA respectively,
show that PR is
perpendicular to QS.

3. Calculate the value of x.

4. The three vertices of triangle ABC lie on a circle. Chords AX, BY, CZ are drawn within the interior angles A, B, C of the triangle. Show that the chords AX, BY and CZ are
the altitudes of triangle XY Z if and only if they are the angle bisectors of triangle ABC.

5. As shown in the diagram, a circle with centre A and radius 9 is tangent to a smaller
circle with centre D and radius
4. Common tangents EF and
BC are
drawn to the circles making points of contact at E, B and C.

Determine the length of EF.

6. In the diagram, two circles are tangent at A and have a common tangent touching
them at B and C respectively.

(a) Show that ∠BAC = 90

^{0}. (Hint: with touching circles it is usual to draw the common tangent at the point of contact!)
(b) If BA is extended to meet the second circle
at D show
that CD is a diameter.

7. If ABCD is a quadrilateral with an inscribed circle as shown, prove
that AB + CD = AD + BC.

8. In this diagram, the two circles are tangent at A. The line BDC is tangent to the smaller circle. Show
that AD bisects
∠BAC.

9. Starting at point A

_{1}on a circle, a particle moves to A_{2}on the circle along chord A_{1}A_{2}which makes a clockwise angle of 35^{0}to the tangent to the circle at A_{1}. From A_{2}the particle moves to A_{3}along chord A_{2}A_{3}which makes a clockwise angle of 37^{0}to the tangent at A_{2}. The particle continues in this way. From A_{k}it moves to A_{k}_{+1}along chord A_{k}A_{k}_{+1}which makes a clockwise angle of (33 + 2k)^{0}to the tangent to the circle at A_{k}. After several trips around the circle, the particle returns to A_{1}for the first time along chord A_{n}A_{1}. Find the value of n.
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