The M1M1 treasure hunt, week 7

At the top of the staircase was a door with a full-length mirror, or looking glass, marked "R²". Passing through this led out onto a plain which stretched as far as Aless could see in all directions. Insects were buzzing everywhere.
"Hello," a passing dragonfly greeted Aless. "We're vectors. Do you need directions?"
"Well, yes please," answered Aless, "Although I don't know where I want to go."
"In that case, you don't have a sensible basis - you may as well follow me." So Aless followed the vector over the open space. After a few miles, they reached two parallel signposts marked "To TweedleLie's house" and "To TweedleBeck's house." Both pointed in the same direction. "I do hope they don't live together," said Aless, "Or they must live an awful long way away."

"Of course they live together," replied the insect. But today they're going to have a battle. In the field over there.
And so saying, the dragonfly recited:

TweedleLie and TweedleBeck agreed to have a fight,
For TweedleLie said TweedleBeck's equations were not right.
"In Algebra we only deal with certainty and proofs;
We leave for Stats and Methods all the vagueness and half-truths."

Aless approached the battle-field, which was being marked out next to a high straight, brick wall, on which sat a strange, egg-shaped creature.
"We have 12m of fencing. The two ends of the fence must be somewhere on my wall. Make a triangle of the largest possible area, using my wall as the third side!" he directed. Aless quickly worked out the largest such triangle had an area

m². ?

TweedleLie was not impressed by the triangular design and called out "That's not good enough - it's not even a ring! For our duel space we should have an ideal domain - we need a field extension!"
Aless sensed there was some more word-play going on, but hadn't yet taken enough modules to understand it fully.
"Very well! Make it into a rectangle!" cried the precariously perched authoritarian figure. A group of workmen (with a single identity, of course) rearranged the fencing and this time, the maximum area was

m². ?

"Is that really the best quadrilateral they can do?" wondered Aless.

TweedleBeck was growing restless. "Try a trapezium!" he trilled triumphantly.
"That's a good idea," thought Aless. "Keep it symmetrical. Suppose they leave the wall at an angle α and build a straight fence of length r. Then they can go parallel to the wall for a distance 12 − 2r and then back symmetrically to the wall. For each α, I can work out the value of r which gives the maximum. Then I can work out which value of α gives the maximum maximum! It's like differentiating with respect to two independent variables! I do hope we'll do that next term."

So, pausing briefly to consider how best to write irrationals such as sqrt(2), Aless duly worked out the largest symmetrical trapezoidal area was

m². ?