Construction of a Triangle from Two Vertices and the Centroid

Jared Brandt has sent me two solutions to the problem of constructing a triangle from two vertices and the centroid. The second one came with a curious commentary. It shows that even after a problem has been solved it pays to have a second look: there may be generalizations or other, simpler solutions.

Hello! I was looking at your page: https://www.cut-the-knot.org/triangle/index.shtml and I have a solution to what seems to be an unanswered problem: {A, B, G}.

It is as follows:

Given: {A, B, G}
Construct: ABC

  1. Draw line AB.
  2. Construct the midpoint of AB, Mc.
  3. Draw the lines McG, AG, and BG.
  4. Construct the perpendicular to AG through B and call the intersection between it and AG, P.
  5. Draw the circle (P, PB) and call the other intersection between it and BP, Q.
  6. Construct the parallel to AG through Q and call the intersection between it and McG, C.

You now have all three point, A, B, and C, and can simply connect the dots to get the triangle!

Thanks for reading!
- Jared

The second solution reached me just a few minutes after the first:

Or you could just be a goof like me and completely overlook the fact that CG = 2McG, and since you know McG at the very beginning, we can quickly find C.

- Jared

Thank you, Jared.


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