Draw a scalene triangle(all sides unequal). On the external side of each side draw an equilateral triangle so that the sides are of the length of that particular side. So if a, b and c are the sides of our triangle then we have 3 equilateral triangles each of a, b and c dimension. Now take the centers of each of these equilateral triangles and join them to form another triangle. Now prove that the triangle so formed is also an equilateral traingle.
Can anyone help?
Thanks
jawahar

Is it not Napoleon's theorem? Check the Glossary for Napoleon.

You will find many proofs there too.

Michael H. Brill

P.S. A related construction that produces a "surprise equilateral triangle" from a scalene triangle is the Morley Trisector Theorem: The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle. For this theorem and its proofs, seee

https://www.cut-the-knot.com/triangle/Morley/Morley.shtml.

A remaining puzzle I have with the Trisector is that you don't have enough angle relationships to solve the problem (the proof of D.J. Newman ASSUMES, but does not derive, enough angles to solve), and yet the construction is done entirely with angles. What's up here?