Triangle $ABC$ is divided into six smaller triangles by lines that pass through the vertices and through a common point inside of the triangle. The areas of four of these triangles are indicated. Calculate the area of triangle $ABC$.
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1. Let the cevians intersect at point \( F \). The cevians divide triangle \( ABC \) into six smaller triangles. We are given the areas of four of these triangles: \( [AFY] = 84 \), \( [CFX] = 70 \), \( [BFX] = 35 \), and \( [BFY] = 65 \).

2. Let the areas of the remaining two triangles be \( x \) and \( y \). Specifically, let \( [AFY] = x \) and \( [CFX] = y \).

3. We know that the ratio of the areas of triangles sharing a common vertex and a common side is equal to the ratio of the lengths of the segments on the opposite side. Therefore, we can set up the following ratios:
   \[
   \frac{x + 84}{y} = \frac{70}{35} = 2
   \]
   This implies:
   \[
   x + 84 = 2y \quad \text{(1)}
   \]

4. Similarly, for the other pair of triangles, we have:
   \[
   \frac{x + 124}{y + 65} = \frac{4}{3}
   \]
   This implies:
   \[
   3(x + 124) = 4(y + 65) \quad \text{(2)}
   \]

5. Simplify equation (2):
   \[
   3x + 372 = 4y + 260
   \]
   \[
   3x - 4y = -112 \quad \text{(3)}
   \]

6. Now we have a system of linear equations:
   \[
   \begin{cases}
   x + 84 = 2y \\
   3x - 4y = -112
   \end{cases}
   \]

7. Solve the first equation for \( x \):
   \[
   x = 2y - 84
   \]

8. Substitute \( x = 2y - 84 \) into the second equation:
   \[
   3(2y - 84) - 4y = -112
   \]
   \[
   6y - 252 - 4y = -112
   \]
   \[
   2y - 252 = -112
   \]
   \[
   2y = 140
   \]
   \[
   y = 70
   \]

9. Substitute \( y = 70 \) back into the equation \( x = 2y - 84 \):
   \[
   x = 2(70) - 84
   \]
   \[
   x = 140 - 84
   \]
   \[
   x = 56
   \]

10. Now we have the areas of all six triangles:
    \[
    [AFY] = 56, \quad [CFX] = 70, \quad [BFX] = 35, \quad [BFY] = 65, \quad [AFY] = 84, \quad [CFX] = 70
    \]

11. The total area of triangle \( ABC \) is the sum of the areas of the six smaller triangles:
    \[
    56 + 84 + 70 + 35 + 65 + 70 = 380
    \]

The final answer is \( \boxed{380} \).