Đang chuẩn bị liên kết để tải về tài liệu:
Collection of all geometric world champion P3

On one of the given lines take segment AB and construct its midpoint, M (cf. Problem 8.74). Let A1 and M1 be the intersection points of lines PA and PM with the second of the given lines, Q the intersection point of lines BM1 and MA1. It is easy to verify that line PQ is parallel to the given lines. In the case when point P does not lie on line AB, we can make use of the solution of Problem 3.36. If point P lies on line AB, then we can first drop perpendiculars l1 and l2 from some other points. | SOLUTIONS 201 8.77. On one of the given lines take segment AB and construct its midpoint M cf. Problem 8.74 . Let A1 and M1 be the intersection points of lines PA and PM with the second of the given lines Q the intersection point of lines BM1 and MA1. It is easy to verify that line PQ is parallel to the given lines. 8.78. In the case when point P does not lie on line AB we can make use of the solution of Problem 3.36. If point P lies on line AB then we can first drop perpendiculars l1 and l2 from some other points and then in accordance with Problem 8.77 draw through point P the line parallel to lines 11 and l2. 8.79. a Let A be the given point l the given line. First let us consider the case when point O does not lie on line l. Let us draw through point O two arbitrary lines that intersect line l at points B and C. By Problem 8.78 in triangle OBC heights to sides OB and OC can be dropped. Let H be their intersection point. Then we can draw line OH perpendicular to l. By Problem 8.78 we can drop the perpendicular from point A to OH. This is the line to be constructed that passes through A and is parallel to l. In order to drop the perpendicular from A to l we have to erect perpendicular l to OH at point O and then drop the perpendicular from A to . If point O lies on line l then by Problem 8.78 we can immediately drop the perpendicular l from point A to line l and then erect the perpendicular to line l from the same point A. b Let l be the given line A the given point on it and BC the given segment. Let us draw through point O lines OD and OE parallel to lines l and BC respectively D and E are the intersection points of these lines with circle S . Let us draw through point C the line parallel to OB to its intersection with line OE at point F and through point F the line parallel to ED to its intersection with OD at point G and finally through point G the line parallel to OA to its intersection with l at point H. Then AH OG OF BC i.e. AH is the segment to be .