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 = 900. (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 A1 on a circle, a particle moves to A2 on the circle along chord A1A2 which makes a clockwise angle of 350 to the tangent to the circle at A1. From A2 the particle moves to A3 along chord A2A3 which makes a clockwise angle of 370 to the tangent at A2. The particle continues in this way. From Ak it moves to Ak+1 along chord AkAk+1 which makes a clockwise angle of (33 + 2k)0 to the tangent to the circle at Ak. After several trips around the circle, the particle returns to A1 for the first time along chord AnA1. Find the value of n.
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