Inscribed Triangle in a Parabola: othogonal projection and mirroring the triangle
I am new to Geogebra and finding it both extraordinary and quite fiddly. I am getting confused trying to do he following on this triangle inscribed in a parabola (screenshot attached. The parabola is x^2 (can I do LaTeX or similar in this forum?). A, B and M are all points on the parabola and they will all move. My task is to determine how to maximise the area of ABM.
The first thing I wanted to do was to have an orthogonal projection of M on AB. I played around with the perpendicular line tools and I am getting all kinds of results except the orthogonal projection. I also tried a segment drawn from M to AV, but of course, the segment does not adjust when I move A, B and/or M. I know this should be simple, but I just cannot work it out.
Another way of approaching the task at hand is to create a parallelogram by mirroring ABM around AB and flipping it horizontally. I used the rotation/translation and mirroring tools but again, I just can't figure out the combination that creates the parallelogram. By mirroring ABM about AB, I get what I think is the right first step, but I don't know how to flip the new triangle obtained to get there (see the second screenshot for where I end up after this first step).
Finally, I have worked out playing around with the points A, B and C that in order for the area of the triangle to be maximised, xm has to be 1/2 (xa+xb). The above are two methods I am trying out to help me reach that conclusion with proof. But is there a way for Geogebra to work out how to maximise the area of ABC wherever A and B are located? I looked at some of the max/maximise functions but I cannot see anything that fits. I also wondered whether that could be done in the CAS perspective but I am just beginning to use Geogebra in earnest and I just don't know how to lay this out in CAS anyway. All suggestions, pointers and comments are very welcome.
Thank you for taking the time to look at this. Any partial answer will help. I am sorry that my knowledge of Geogebra is so basic that I have to ask so many questions: thank you for your patience.