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A, B, C, D, E ........ Z are the points marked on the circumference of a circle equidistantly. What can be the maximum number of triangles which can be formed using three points as vertices such that their circumcentre lies on one of the sides of a triangle?
  • a)
    24
  • b)
    13
  • c)
    372
  • d)
    312
Correct answer is option 'D'. Can you explain this answer?
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A, B, C, D, E ........ Z are the points marked on the circumference of...
Maximum number of triangles with circumcentre on a side
To find the maximum number of triangles that can be formed using three points as vertices such that their circumcentre lies on one of the sides of a triangle, we need to consider the possible cases and combinations.

Understanding the concept
- The circumcentre of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect.
- Since the points A, B, C, D, E...Z are equidistantly marked on the circumference of the circle, the circumcentre will lie on one of the sides of the triangle formed by any three points.

Determining the maximum number of triangles
- We can choose any 3 points out of 26 points in C(26,3) ways.
- For each set of 3 points, the circumcentre can lie on one of the sides of the triangle.
- So, the total number of triangles with circumcentre on a side is C(26,3) = 26! / (3! * 23!) = 26 * 25 * 24 / 6 = 312.
Therefore, the maximum number of triangles that can be formed using three points as vertices such that their circumcentre lies on one of the sides of a triangle is 312, which corresponds to option 'D'.
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