# Triangle Square Inscribing

I'm reading van Lamoen, Floor and Weisstein's article Triangle Square Inscribing from the MathWorld site and reached some hurdles I can't jump over. Specifically,

1. In the first sentence, they mention two types of squares inscribing a triangle--type I has "two adjacent vertices of the square on one side" and type II has "two opposite vertices on one side." I can't come up with an example of a type II.

2. In the next paragraph, it is written, "These squares, however, are not necessarily the largest inscribed squares." I don't know what is meant by largest in this context. Ratio of the square's area to triangle's area?

3. In that same paragraph but a few lines earlier, one of the steps in the construction says to "construct $FK$". Why? It seems irrelevant--or am I overlooking something?

4. Fourth paragraph from the bottom, we read, "A similar construction can be done by initially erecting a square internally on the side $BC$." Does this method produce an triangle inscribed square? I'm having trouble reproducing the steps of the first construction with an internally erected square and the figures don't help at all. (For one, they're too small.)

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They are not looking at inscribed squares, but at squares whose vertices are on the lines (not necessarily line segments) determined by the vertices of the triangle. – André Nicolas Dec 17 '11 at 20:32