Question 1: Make CD perpendicular to AB at point C, and then prove that triangle MEA is equal to triangle DAC and triangle NFB is equal to triangle BDC(K congruence).

The second question: When the triangle ABC is an isosceles triangle, the area is the largest. The maximum area is: 1*(0.5+ half of root number 2) *0.5=( 1+ root number 2)/4.

The third question: it is a general's horse drinking water problem (Jianshui pumping station problem), and the answer is: under the root number (7+4* root number 2).