Any help on these questions would be great, they're the only five I couldn't get my head around out of a set of 32!
1) P is a midpoint of a chord AB of a circle. XY is a chord containing the point P. The tangents at X and Y meet AB extended at M and N respectively. Prove that AM = BN.
2) PQ and RS are common tangents to two intersecting circles. If T is one of the points of intersection of the two circles, prove that the circles through T, P, Q and through T, R, S touch each other.
3) ABC is a triangle inscribed in a circle. BE and CF are perpendiculars to AC and AB respectively. AP is drawn perpendicular to EF and, when produced meets the circle at Q. Prove that angle ABQ is a right angle.
4) Draw three circles that intersect the other two. Prove that the three common chords are concurrent.
5) A right angled triangle ABC with right angle at A circumscribes a circle of radius r. Prove that r = 0.5(c + b - a) where a, b, c are the length measures of the sides of the triangle.