1. **Problem statement:** We have a large equilateral triangle $ABC$ and a smaller inverted equilateral triangle $DEF$ inside it. Points $D$, $E$, and $F$ lie on sides $AB$, $BC$, and $AC$ respectively, dividing the original triangle into four smaller triangles. 2. **Goal:** Understand the configuration and properties of the smaller triangle $DEF$ inside $ABC$. 3. **Step 1: Properties of equilateral triangles** - All sides are equal. - All angles are $60^\circ$. 4. **Step 2: Position of points $D$, $E$, and $F$** - $D$ lies on $AB$. - $E$ lies on $BC$. - $F$ lies on $AC$. 5. **Step 3: Inverted equilateral triangle $DEF$** - Since $DEF$ is equilateral and inverted, it is rotated $180^\circ$ relative to $ABC$. - The points $D$, $E$, and $F$ divide the sides of $ABC$ such that $DEF$ is similar to $ABC$ but smaller and upside down. 6. **Step 4: Division into four smaller triangles** - The original triangle $ABC$ is divided into four smaller triangles: $DEF$ and three others adjacent to each side. 7. **Step 5: Ratios and lengths** - If the side length of $ABC$ is $s$, and the smaller triangle $DEF$ has side length $t$, then the ratio $\frac{t}{s}$ depends on the exact positions of $D$, $E$, and $F$. 8. **Step 6: Common case - medial triangle** - If $D$, $E$, and $F$ are midpoints of $AB$, $BC$, and $AC$, then $DEF$ is the medial triangle. - The medial triangle is similar to $ABC$ with side length $\frac{s}{2}$. - The area of $DEF$ is $\frac{1}{4}$ of $ABC$. 9. **Step 7: General case** - Without specific lengths or ratios, we cannot compute exact values. **Final answer:** The smaller inverted equilateral triangle $DEF$ inside $ABC$ divides the original triangle into four smaller triangles. If $D$, $E$, and $F$ are midpoints, $DEF$ is the medial triangle with side length half of $ABC$ and area one quarter of $ABC$. Otherwise, the exact sizes depend on the positions of $D$, $E$, and $F$.