We are tasked with finding the area of triangle $MAB$, where:

• $ABCD$ is a square with side length $16$,
• $E$ and $F$ are points on side $CD$ such that $CE = EF = FD$, i.e., the points divide $CD$ into three equal segments,
• $BF$ and $AE$ are lines that intersect at point $M$.

Step 1: Coordinates of key points

• Let the square $ABCD$ be on the coordinate plane with $A = (0, 16)$, $B = (16, 16)$, $C = (16, 0)$, and $D = (0, 0)$.
• Since $E$ and $F$ divide $CD$ into three equal parts, we have:
  • $E = \left( \frac{2}{3} \times 16, 0 \right) = (10.67, 0)$,
  • $F = \left( \frac{1}{3} \times 16, 0 \right) = (5.33, 0)$.

Step 2: Equations of lines $AE$ and $BF$

• The slope of line $AE$ (connecting $A = (0, 16)$ and $E = (10.67, 0)$) is: $\text{slope of } AE = \frac{0 - 16}{10.67 - 0} = -\frac{16}{10.67} \approx -1.5$ So, the equation of line $AE$ is: $y - 16 = -1.5(x - 0) \quad \Rightarrow \quad y = -1.5x + 16$
• The slope of line $BF$ (connecting $B = (16, 16)$ and $F = (5.33, 0)$) is: $\text{slope of } BF = \frac{0 - 16}{5.33 - 16} = \frac{-16}{-10.67} \approx 1.5$ So, the equation of line $BF$ is: $y - 16 = 1.5(x - 16) \quad \Rightarrow \quad y = 1.5x - 8$

Step 3: Finding the intersection point $M$

To find the coordinates of point $M$, we set the equations of $AE$ and $BF$ equal: $-1.5x + 16 = 1.5x - 8$ Solving for $x$: $16 + 8 = 1.5x + 1.5x \quad \Rightarrow \quad 24 = 3x \quad \Rightarrow \quad x = 8$ Substitute $x = 8$ into the equation of $AE$ to find $y$: $y = -1.5(8) + 16 = -12 + 16 = 4$ So, $M = (8, 4)$.

Step 4: Area of triangle $MAB$

The area of triangle $MAB$ can be found using the formula for the area of a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$: $\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$ Substituting the coordinates $A(0, 16)$, $B(16, 16)$, and $M(8, 4)$: $\text{Area} = \frac{1}{2} \left| 0(16 - 4) + 16(4 - 16) + 8(16 - 16) \right|$ $= \frac{1}{2} \left| 0 + 16(-12) + 8(0) \right| = \frac{1}{2} \left| -192 \right| = \frac{192}{2} = 96$

Thus, the area of triangle $MAB$ is $\boxed{96}$.

Do you have any questions or want details on a specific part of this solution? Here are 5 related questions:

1. How does the division of $CD$ into equal parts affect the result?
2. Can we find the area of triangle $MAB$ using a different method?
3. What is the geometric significance of the intersection of $AE$ and $BF$?
4. How would the result change if the square had a different side length?
5. How can the coordinates of intersection points be applied to other geometric problems?

Tip: When finding the area of a triangle from its vertex coordinates, always verify the orientation of the points to avoid sign errors in the formula.