By I. F. Sharygin
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Additional resources for Problems in Plane Geometry (Science for Everyone)
What are the limits of change of the distance between the centre of the inscribed circle and the point of intersection of the medians? 160. The sides of a parallelogram are equal to a and b (a =1= b). What are the limits of change of the cosine of the acute angle between the diagonals? 161. Three straight lines are drawn through a point M inside a triangle ABC parallel to its sides. The segments of the lines enclosed inside the triangle are equal to one another. Find their length if the sides of the triangle are a, b, and c.
A straight line passing through the point M intersects the sides of the angle and cuts off a triangle whose perimeter is 12. Find the area of this triangle. 151. Given a rectangle ABeD in which I AB I = 4, I Be I = 3. Find the side of the rhombus one vertex of which coincides with A, and three others lie on the line segments AB, BC and BD (one vertex on each segment). 152. Given a square ABCD with a side equal to 1. Find the side of the rhombus one vertex of which coincides with A, the oppo- va 3-01557 va 34 'Problems in Plane Geometry site vertex lies on the line BD, and the two remaining vertices on the lines BC and CD.
187. The bisector of each angle of a triangle intersects the opposite side at a point equidistant from the midpoints of the two other sides of the triangle. Does it, in fact, mean that the triangle is regular? 188. Given in a triangle are two sides: a and b (a > b). Find the third side if it is known that a ha ~ b h b , where h a and h b are the altitudes dropped on these sides (h a the altitude drawn to the side a). 189. Given a convex quadrilateral ABG'D circumscribed about a circle of diameter 1.
Problems in Plane Geometry (Science for Everyone) by I. F. Sharygin