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Archive for Tháng Mười, 2011

On the figure of IMO 2009 P2 Generalization

For the generalization, see the file below:

IMO P2 generalization

We found some nice properties about this figure.

Problem 1: Given two triangles ABC  and XYZ with their circumcircle (O). YZ intersects AB, AC at P, N, respectively. Let J be the projection of O on YZ. Let K, L be the midpoints of BN, CP. A' be the reflection of J wrt KL. Similarly we define B', C'. Prove that:

1 (Nguyen Van Thanh).  A' lies on the 9-point circle of triangle ABC.

2.  Two triangles XYZ and A'B'C' are similar.

Problem 2: Given a triangle ABC and its circumcircle (O). Let d be an arbitrary line.  d intersects BC, CA, AB at X, Y, Z. Let L be the projection of O on d. Prove that (ALX), (BLY), (CLZ) are coaxal.

Problem 3:  Given triangle ABC, its circumcircle (O) and a vector \vec{u} . Let d be a line such that d // \vec{u}. d intersects AB, AC at Y, Z. Let M, N be the midpoints of  BY, CZ, respectively; L be the projection of O on d. Prove that when d moves, the line through L and perpendicular to MN passes through a fixed point.

Have fun! 🙂

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