1. **Problem Statement:** We have triangle $ABC$ with points $D$, $E$, and $F$ as midpoints of segments $AC$, $BD$, and $AE$ respectively. Given the area of triangle $BEF$ is 5, we need to find the area of triangle $ABC$. 2. **Key Concept:** When a point is the midpoint of a segment, it divides the segment into two equal parts. Also, the area of a triangle formed by midpoints of sides of another triangle is related by specific ratios. 3. **Step 1: Understand the points** - $D$ is midpoint of $AC$. - $E$ is midpoint of $BD$. - $F$ is midpoint of $AE$. 4. **Step 2: Area relations** - Since $D$ is midpoint of $AC$, triangle $BDC$ shares half the base $AC$ compared to $ABC$. - $E$ midpoint of $BD$ means $E$ divides $BD$ into two equal parts. - $F$ midpoint of $AE$ means $F$ divides $AE$ into two equal parts. 5. **Step 3: Area of triangle $BEF$ in terms of $ABC$** - Each midpoint division reduces the area by a factor of $\frac{1}{2}$ in one dimension. - Triangle $BEF$ is formed by midpoints on segments inside $ABC$, so its area is a fraction of $ABC$. 6. **Step 4: Calculate area ratio** - Area of $BDC$ is $\frac{1}{2}$ area of $ABC$ because $D$ is midpoint of $AC$. - $E$ is midpoint of $BD$, so segment $BE$ is half of $BD$. - $F$ is midpoint of $AE$, so segment $AF$ is half of $AE$. 7. **Step 5: Area of $BEF$ relative to $BDC$** - Triangle $BEF$ is inside $BDC$ with vertices at midpoints, so its area is $\frac{1}{4}$ area of $BDC$. 8. **Step 6: Combine ratios** - Area($BEF$) = $\frac{1}{4}$ Area($BDC$) = $\frac{1}{4} \times \frac{1}{2}$ Area($ABC$) = $\frac{1}{8}$ Area($ABC$). 9. **Step 7: Solve for Area($ABC$)** $$\text{Area}(BEF) = \frac{1}{8} \text{Area}(ABC)$$ $$5 = \frac{1}{8} \text{Area}(ABC)$$ $$\text{Area}(ABC) = 5 \times 8 = 40$$ **Final answer:** The area of triangle $ABC$ is 40.