Triangle In The Square

First, recall that the area of a triangle is $\dfrac 12 bh$ where $b$ denotes the length of the base and $h$ the height. Keeping $h$ fixed, we may increase $b$ to increase the area of the triangle. Similarly, keeping $b$ fixed, we may increase $h$ to increase the area of the triangle. This shows that the vertices must lie along the boundary of the square.

Now, we claim that one vertex lies at the corner of the square. Suppose not. Then, pick any edge of the triangle. Consider the vertex not along this edge. By moving this vertex away from the edge, we may increase the height of the triangle, by the remarks above. Since a corner of the square will be the furthest distance from our selected edge, moving the opposite vertex to the corner of the square maximizes the area.

Now, let $S$ be a square in the plane. Let $s$ denote the length of the edges of $S$ . Without loss of generality, we may suppose that $S$ lies in the first quadrant with one vertex at the origin. Consider a triangle in $S$ that maximizes area. By the above remarks, we may suppose that one vertex of the triangle lies at the origin.

By the above remarks on the construction of the triangle, $b=s$ . Similarly, $h=s$ . So the area of the triangle is $\dfrac 12 bh=\dfrac 12 s^2=\dfrac 12 Ar($ Square $)$ . (You may want to argue these points more carefully, but they can be argued by using the ideas of the first two paragraphs.)

Note that the second statement is not true: Let the vertices of the sqaure be $(0,0),(1,0),(0,1),$ and $(1,1)$ . Let the vertices of the triangle be $(0,0),(1,0)$
and $(\dfrac 12 ,1)$ . Then we still have $Ar(△)=\dfrac 12 Ar(□).$