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First construct a triangle.

Then construct the midpoints of the sides of this triangle.

Now join these points

Now construct the circumcentre of this smaller triangle.

This is the nine-point circle of the triangle ABC. To construct the other 6 points, you need to find the altitudes of ABC. The feet of these altitudes gives you the first 3 points.

The altitudes intersect in the orthocentre of ABC. Midpoints of the lines which connect orthocentre with the vertices A, B, and C, will give you further 3 points.

or about various properties of triangles:

Euler Line

Circumcentre

Centroid

Orthocentre

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