Let $ABC$ be a triangle. Find all the triangles $XYZ$ with vertices inside triangle $ABC$ such that $XY,YZ,ZX$ and six non-intersecting segments from the following $AX, AY, AZ, BX, BY, BZ, CX, CY, CZ$ divide the triangle $ABC$ into seven regions with equal areas.
1. **Shift the sides inward**: First, we need to shift all three sides of triangle \(ABC\) inward by \(\frac{1}{7}\) of the corresponding altitude. This will create a smaller triangle \(MNP\) inside \(ABC\). The vertices \(M\), \(N\), and \(P\) are located on the sides \(BC\), \(CA\), and \(AB\) respectively.

2. **Determine the positions of \(X\), \(Y\), and \(Z\)**: The points \(X\), \(Y\), and \(Z\) must lie on the sides of triangle \(MNP\). Specifically, \(X\) lies on \(MN\), \(Y\) on \(NP\), and \(Z\) on \(PM\).

3. **Calculate the ratios**: To ensure that the triangle \(XYZ\) divides the triangle \(ABC\) into seven regions of equal area, we need to find the correct ratios for the segments:
   - Let \(AX = k \cdot BX\)
   - Let \(BY = m \cdot CY\)
   - Let \(CZ = n \cdot AZ\)

4. **Equal area condition**: The condition that the areas of the seven regions are equal implies that the areas of triangles \(AXZ\), \(BXY\), and \(CYZ\) must each be \(\frac{1}{7}\) of the area of triangle \(ABC\).

5. **Using coordinates**: To verify the areas, we can use the coordinate geometry approach. Assume \(A = (0, 0)\), \(B = (1, 0)\), and \(C = (0, 1)\). The vertices of triangle \(MNP\) can be calculated as follows:
   - \(M\) is \(\frac{1}{7}\) of the way from \(B\) to \(C\), so \(M = \left( \frac{6}{7}, \frac{1}{7} \right)\)
   - \(N\) is \(\frac{1}{7}\) of the way from \(C\) to \(A\), so \(N = \left( 0, \frac{6}{7} \right)\)
   - \(P\) is \(\frac{1}{7}\) of the way from \(A\) to \(B\), so \(P = \left( \frac{1}{7}, 0 \right)\)

6. **Finding \(X\), \(Y\), and \(Z\)**: The points \(X\), \(Y\), and \(Z\) must split the sides of \(MNP\) such that the areas of the triangles \(AXZ\), \(BXY\), and \(CYZ\) are equal. This requires:
   - \(BX = 3AX\)
   - \(CY = 3BY\)
   - \(AZ = 3CZ\)

7. **Verification**: To verify, we can calculate the areas of the triangles using the determinant formula for the area of a triangle given its vertices. For example, the area of triangle \(AXZ\) can be calculated as:
   \[
   \text{Area}(AXZ) = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
   \]
   where \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the coordinates of \(A\), \(X\), and \(Z\) respectively.

8. **Conclusion**: By ensuring that the points \(X\), \(Y\), and \(Z\) split the sides of \(MNP\) in the specified ratios, we can confirm that the areas of the seven regions are equal.

\(\blacksquare\)

The final answer is \( \boxed{ BX = 3AX } \), \(CY = 3BY\), and \(AZ = 3CZ\).