1. A circle centered on side *AC* of triangle *ABC* passes through vertex *C* and touches line *AB* at point *B.* Find the diameter of the circle if *AB* = 15, *AC* = 25. Answer: 16

2. A circle inscribed in triangle ABC touches its sides at points M, K and P. Find the angles of triangle ABC if the angles of triangle MKP are 49°, 69° and 62°. Answer: 82°, 42°, 56°

3. Two tangents are drawn from point A to a circle centered at point O. Find the radius equal to 8. Answer: 4

4. The circle intersects sides AB and AC of triangle ABC at points K and P, respectively, and passes through vertices B and C. Find the length of segment KP if AK = 18 and side AC is 1.2 times longer than side BC. Answer:15

5. Segments AB and DC lie on parallel lines, and segments AC and BD intersect at point M. Find MC if AB = 16, DC = 24 , AC = 25. Answer: 15

6. In triangle *ABC* , angles *A* and *C* are 40° and 60° respectively. Find the angle between height *BH* and bisector *BD* . Answer: 10

7. The segments *AB* and *DC* lie on parallel lines, and the segments *AC* and *BD* intersect at the point *M.* Find *MC* if *AB* =10, *DC* =25, *AC* =56. Answer: 40

8. Bisectors of angles *A* and *D* of parallelogram *ABCD* intersect at a point lying on side *BC* . Find *BC* if *AB* = 34. Answer: 68

9. Straight, parallel to the bases and trapeze , passes through the point of intersection of the diagonals of the trapezoid and intersects its sides and at points and respectively. Find the length of the segment , if , . Answer: 12

10. In a right triangle ABC with a right angle C, the legs are known: AC=6, BC=8. Find the median SC of this triangle. Answer:5

11. The circle passes through the vertices A and C of the triangle ABC and intersects its sides AB and BC at points K and E, respectively. Segments AE and SK are perpendicular. Find ∠SWR if ∠ABC = 20°. Answer: 35

12. In triangle ABC, angles A and C are 20° and 60° respectively. Find the angle between height BH and bisector BD. Answer: 20

13. Line AD, perpendicular to median BM of triangle ABC, bisects it. Find side AC if side AB is 4. Answer: 8

14. The leg and hypotenuse of a right triangle are 18 and 30. Find the height drawn to the hypotenuse. Answer: 14.4

15. Point H is the base of height BH drawn from the vertex of right angle B of right triangle ABC. A circle with diameter BH intersects sides AB and CB at points P and K, respectively. Find PK if BH = 16 Answer: 16

16. The circle intersects sides AB and AC of triangle ABC at points K and P, respectively, and passes through vertices B and C. Find the length of segment KP if AP = 18 and side BC is 1.2 times less than side AB. Answer:15

17. A line parallel to side AC of triangle ABC intersects sides AB and BC at points M and N, respectively. Find BN if MN = 13, AC = 65, NC = 28 Answer: 7

18. Find the ratio of two sides of a triangle if its median, coming out of their common vertex, forms angles of 30° and 90° with these sides. Answer: 1:2

19. The height of a triangle divides its base into two segments with lengths 8 and 9. Find the length of this height if it is known that another height of the triangle bisects it.

Answer: 12

20. The bisectors of angles A and B at the lateral side AB of the trapezoid ABCD intersect at point F. Find AB if AF=24. BF=10 . Answer: 26

21. The diagonals AC and VD of the trapezoid ABCD intersect at the point O. The areas of the triangles AOD and BOS are 16 and 9, respectively. Find the area of the trapezoid. Answer: 49

22. In trapezoid ABCD, the base of AD is twice the base of BC and twice the side of CD. Angle ADC is 60°, side AB is 2. Find the area of the trapezoid. Answer:

23. The bases of the trapezoid are 16 and 34. Find the line segment connecting the midpoints of the diagonals of the trapezoid. Answer: 9

24. Bisectors of angles *A* and *D* of parallelogram *ABCD* intersect at a point lying on side *BC* . Find *AB* if *BC* = 34 Answer: 17

25. Bisectors of angles A and B of parallelogram ABCD intersect at point K. Find the area of the parallelogram if BC = 19 and the distance from point K to side AB is 7. Answer: 266

26. Find the side AB of the trapezoid ABCD if the angles ABC and BCD are 30° and 120° respectively, and CD = 25. Answer:

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