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In the triangle ABC let B' and C' be the midpoints of the sides AC and AB respectively and H the foot of the altitude passing through the vertex A. Prove that the circumcircles of the triangles AB'C',BC'H, and B'CH have a common point I and that the line HI passes through the midpoint of the segment B'C'.

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A point M is chosen on the side AC of the triangle ABC in such a way that the radii of the circles inscribed in the triangles ABM and BMC are equal. Prove that
BM^{2} = X \cot \left( \frac {B}{2}\right)
where X is the area of triangle ABC.
Let Q be the centre of the inscribed circle of a triangle ABC. Prove that for any point P,
a(PA)^2 + b(PB)^2 + c(PC)^2 = a(QA)^2 + b(QB)^2 + c(QC)^2 + (a + b + c)(QP)^2,
where a = BC, b = CA and c = AB.
Let ABC be an acute-angled triangle. The lines L_{A}, L_{B} and L_{C} are constructed through the vertices A, B and C respectively according the following prescription: Let H be the foot of the altitude drawn from the vertex A to the side BC; let S_{A} be the circle with diameter AH; let S_{A} meet the sides AB and AC at M and N respectively, where M and N are distinct from A; then let L_{A} be the line through A perpendicular to MN. The lines L_{B} and L_{C} are constructed similarly. Prove that the lines L_{A}, L_{B} and L_{C} are concurrent.
The triangle ABC is inscribed in a circle. The interior bisectors of the angles A,B and C meet the circle again at A', B' and C' respectively. Prove that the area of triangle A'B'C' is greater than or equal to the area of triangle ABC.
Let ABCD be a convex quadrilateral whose vertices do not lie on a circle. Let A'B'C'D' be a quadrangle such that A',B', C',D' are the centers of the circumcircles of triangles BCD,ACD,ABD, and ABC. We write T (ABCD) = A'B'C'D'. Let us define A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).

(a) Prove that ABCD and A''B''C''D'' are similar.

(b) The ratio of similitude depends on the size of the angles of ABCD. Determine this ratio.
The circle inscribed in a triangle ABC touches the sides BC,CA,AB in D,E, F, respectively, and X, Y,Z are the midpoints of EF, FD,DE, respectively. Prove that the centers of the inscribed circle and of the circles around XYZ and ABC are collinear.