Question:

Step 1: Identify the relevant circumcircles. Let $\mathcal{C}_1$ be the circumcircle of triangle $BDC$. This circle passes through points $B$, $D$, and $C$. It meets side $AB$ at point $E$. Let $\mathcal{C}_2$ be the circumcircle of triangle $ABD$. This circle passes through points $A$, $B$, and $D$. It meets side $BC$ at point $F$. Step 2: Apply the Power of a Point Theorem for point $A$ with respect to $\mathcal{C}_1$. Point $A$ lies outside $\mathcal{C}_1$. The line segment $AEC$ is a secant to $\mathcal{C}_1$ passing through $A$, $E$, and $B$. The line segment $ADC$ is another secant to $\mathcal{C}_1$ passing through $A$, $D$, and $C$. By the Power of a Point Theorem, the product of the lengths of the segments from $A$ to the circle along each secant is equal: $$ AE \cdot AB = AD \cdot AC $$ From this, we can express $AE$ as: $$ AE = \frac{AD \cdot AC}{AB} $$ Step 3: Apply the Power of a Point Theorem for point $C$ with respect to $\mathcal{C}_2$. Point $C$ lies outside $\mathcal{C}_2$. The line segment $CFB$ is a secant to $\mathcal{C}_2$ passing through $C$, $F$, and $B$. The line segment $CDA$ is another secant to $\mathcal{C}_2$ passing through $C$, $D$, and $A$. By the Power of a Point Theorem, the product of the lengths of the segments from $C$ to the circle along each secant is equal: $$ CF \cdot CB = CD \cdot CA $$ From this, we can express $CF$ as: $$ CF = \frac{CD \cdot CA}{CB} $$ Step 4: Formulate a ratio of $AE$ to $CF$. Now we take the ratio of the expressions for $AE$ and $CF$: $$ \frac{AE}{CF} = \frac{\frac{AD \cdot AC}{AB}}{\frac{CD \cdot CA}{CB}} $$ $$ \frac{AE}{CF} = \frac{AD \cdot AC}{AB} \cdot \frac{CB}{CD \cdot CA} $$ We can cancel out the common term $AC$: $$ \frac{AE}{CF} = \frac{AD \cdot CB}{AB \cdot CD} $$ Step 5: Apply the Angle Bisector Theorem. Since $BD$ is the internal angle bisector of angle $B$ in triangle $ABC$, according to the Angle Bisector Theorem, the ratio of the sides adjacent to the bisected angle is equal to the ratio of the segments the bisector divides the opposite side into: $$ \frac{AD}{CD} = \frac{AB}{BC} $$ Rearranging this equation, we get: $$ AD \cdot BC = AB \cdot CD $$ Substitute this relationship into the ratio of $AE$ to $CF$ found in Step 4: $$ \frac{AE}{CF} = \frac{AD \cdot CB}{AB \cdot CD} = \frac{(AB \cdot CD)}{AB \cdot CD} $$ $$ \frac{AE}{CF} = 1 $$ This implies that $AE = CF$. Step 6: Determine the final value of $CF$. We are given that $AE = 3$. Since we have established that $AE = CF$, it follows that $CF = 3$. The final answer is $3$.