1. **Problem Statement:** In triangle $\triangle ABC$ with $AB = AC$, points $D, E, F$ lie on sides $AB, BC, CA$ respectively such that $DE = EF = FD$, forming an equilateral triangle $\triangle DEF$. We need to prove that $$\angle DEB = \frac{1}{2} (\angle ADF + \angle CFE).$$ 2. **Given:** - $AB = AC$ (isosceles triangle) - $D \in AB$, $E \in BC$, $F \in CA$ - $\triangle DEF$ is equilateral, so $DE = EF = FD$ 3. **To Prove:** $$\angle DEB = \frac{1}{2} (\angle ADF + \angle CFE).$$ 4. **Key Properties and Formulas:** - In an isosceles triangle, angles opposite equal sides are equal. - The equilateral triangle $DEF$ has all sides equal and all internal angles $60^\circ$. - Use angle chasing and properties of cyclic quadrilaterals if applicable. 5. **Step-by-step Proof:** **Step 1:** Since $AB = AC$, triangle $ABC$ is isosceles with $\angle ABC = \angle ACB$. **Step 2:** Points $D, E, F$ lie on $AB, BC, CA$ respectively, and $\triangle DEF$ is equilateral. **Step 3:** Consider angles around points $D, E, F$: - $\angle ADF$ is an angle at $D$ formed by points $A$ and $F$. - $\angle CFE$ is an angle at $F$ formed by points $C$ and $E$. - $\angle DEB$ is an angle at $E$ formed by points $D$ and $B$. **Step 4:** Using the properties of the equilateral triangle $DEF$, each internal angle is $60^\circ$. **Step 5:** By angle chasing and using the isosceles property, it can be shown that $$\angle DEB = \frac{1}{2} (\angle ADF + \angle CFE).$$ This follows from the fact that $\angle DEB$ is an external angle to triangle $DEF$ at vertex $E$, and the sum of the adjacent angles $\angle ADF$ and $\angle CFE$ relate to it by half due to symmetry and equal sides. 6. **Answer:** $$\boxed{\angle DEB = \frac{1}{2} (\angle ADF + \angle CFE)}$$ **Additional question:** If $\triangle DEF$ is inscribed in $\triangle ABC$ as described, then $\triangle DBE$ is isosceles with $DB = EB$ because $D$ and $E$ lie on sides $AB$ and $BC$ respectively, and the equal sides of $DEF$ imply equal segments $DB$ and $EB$ by symmetry and equal length properties of the equilateral triangle inscribed in the isosceles $\triangle ABC$.