By I. F. Sharygin
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Example text
One of the segments of the side from the vertex to the point of tangency is equal to V Find the area of the parallelogram. 130. A circle of radius R passes through the vertices A and B of the triangle ABC and touches the line A C at A. Find the area of the triangle ABC if LB = a, LA = a ~. 131. In a triangle ABC, the angle bisector AK is perpendicular to the median BM, and the angle B is equal to 120°. Find the ratio of the area of the triangle A-BC to the area of the circle circumscribed about this triangle.
A point C is on the arc AB such that LAOe = 60° (LBOC = 30°). A circle of radius AB centred at A intersects the extension of OC beyond the point C at D. Prove that the line segment CD is equal to the side of a regular decagon inscribed in the circle. Let us now take a point M diametrically opposite to the point C. The line segment MD, increased by 1/5 of its length, is assumed to be approximately equal to half the circumference. Estimate the error of this approximation. 144. Given a rectangle 7 X 8.
Find I MN I· 232. In a triangle ABC, the angle B is 'Jt/4, the angle C is :1/6. Constructed on the medians BN and CN as diameters are circles intersecting each other at points P and Q. The chord PQ intersects the side BC at a point D. Find the ratio I BD I IDC I. 233. Let AB denote the diameter of a circle, 0 its centre, AB = 2R, C a point on the circle, M a point on the chord AC. From the point M, a perpendicular MN is dropped on AB and another one is erected to AC intersecting the circle at L (the line segment CL intersects AB).