Geometry problems of the week!

Dear all members, please try to solve these exercises, we will be able to improve your geometry skill! Try to think it one by one, consider it carefully... You may be able to discuss these problems at www.groups.google.com/group/groupcmg .... Enjoy!

1.(Chinese Mathematical Olympiad, 2007) Triangle ABC is not isosceles. The incenter is I, the excenter is O. The incircle touches the sides BC, CA, AB at points D, E, F, respectively. Lines FD and AC intersect at P, lines DE and AB intersect at Q. The midpoints of segments EP amd FQ are M and N, respectively. Prove that MN and OI are perpendicular.

2. Determine the locus of the centres of all regular triangles circumscribed about a given acute triangle. (The triangle G is said to be circumscribed about the triangle H if the vertices of H lie on the sides of G.)

3. Given a convex quadrilateral ABCD and a point P in its interior such that AP=CP, and. Prove that DA.AB.BP=BC.CD.DP.

4. The angle bisector drawn from vertex C of an acute angled triangle ABC intersects the opposite side at the point F. The feet of the perpendiculars drawn from the point F to the sides BC and CA are P and Q, respectively. Let M denote the intersection of the lines AP and BQ. Prove that AB and CM are perpendicular to each other.

5. A' is the reflection of the vertex A of an equilateral triangle ABC about the opposite side. A line passing through A' intersects the lines AB and AC at the points C' and B', respectively. What is the locus of the intersection of lines BB' and CC'?

6. The angle bisectors of triangle ABC intersect sides BC, CA and AB at points A1, B1 and C1, respectively. On line A1B1, denote by F the perpendicular foot point of C1. Prove that line FC1 bisects angle AFB.

7. Let h denote the length of the tangents drawn to a circle from an exterior point P, and let the midpoint of the line segment connecting the points of contact be F. Prove that a chord AB of the circle satisfies the equality AP.PB=h2 if and only if the line AB passes through the point P or the point F.

8. The angle bisector drawn from vertex C of an acute angled triangle ABC intersects the opposite side at the point F. The feet of the perpendiculars drawn from the point F to the sides BC and CA are P and Q, respectively. Let M denote the intersection of the lines AP and BQ. Prove that AB and CM are perpendicular to each other.