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

Draw line AB.
Construct the midpoint of AB, M_c.
Draw the lines M_cG, AG, and BG.
Construct the perpendicular to AG through B and call the intersection between it and AG, P.
Draw the circle (P, PB) and call the other intersection between it and BP, Q.
Construct the parallel to AG through Q and call the intersection between it and M_cG, 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 = 2M_cG, and since you know M_cG at the very beginning, we can quickly find C.

- Jared

Thank you, Jared.

71546315