By Viktor Prasolov, Dimitry Leites (translator)

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**Extra info for Problems in Plane and Solid Geometry, Volume 1, Plane Geometry **

**Sample text**

46. Diagonals AC and CE of a regular hexagon ABCDEF are divided by points M and N , respectively, so that AM : AC = CN : CE = λ. Find λ if it is known that points B, M and N lie on a line. 47. The corresponding sides of triangles ABC and A1 B1 C1 are parallel and sides AB and A1 B1 lie on one line. Prove that the line that connects the intersection points of the circumscribed circles of triangles A1 BC and AB1 C contains point C1 . 48. In triangle ABC heights AA1 , BB1 and CC1 are drawn. Line KL is parallel to CC1 ; points K and L lie on lines BC and B1 C1 , respectively.

Prove that points A, C, O and Ob lie on a circle centered at M . b) Point O inside triangle ABC is such that lines AO, BO and CO pass through the centers of the circumscribed circles of triangles BCO, ACO and ABO, respectively. Prove that O is the center of the inscribed circle of triangle ABC. 5. Vertices A and B of right triangle ABC with right angle ∠C slide along the sides of a right angle with vertex P . Prove that in doing so point C moves along a line segment. 6. Diagonal AC of square ABCD coincides with the hypothenuse of right triangle ACK, so that points B and K lie on one side of line AC.

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