Originally Posted by

**AlvinCY** 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 s= 0.5(c + b - a)* where *a, b, c* are the length measures of the sides of the triangle.