In triangle ABC, points D, E, F are on sides BC, CA, AB respectively such that AD, BE, CF are concurrent. If BD:DC = 2:3, CE:EA = 3:4, find AF:FB using Ceva's theorem.

Step 1: State Ceva's Theorem

Ceva's Theorem states that for a triangle ABC, if points D, E, F are on sides BC, CA, AB respectively, then the lines AD, BE, CF are concurrent if and only if the product of the ratios of the segments on each side is equal to 1. This theorem provides a powerful tool for solving problems involving concurrent lines in a triangle.

Step 2: Substitute Given Ratios

We are given the ratios $BD:DC = 2:3$ and $CE:EA = 3:4$. We substitute these values into Ceva's Theorem. The ratio $BD:DC$ can be written as $\frac{BD}{DC} = \frac{2}{3}$, and $CE:EA$ as $\frac{CE}{EA} = \frac{3}{4}$.

Step 3: Simplify the Equation

Now, we simplify the product of the known ratios. Multiply the numerators and denominators: $2 \cdot 3 = 6$ and $3 \cdot 4 = 12$. This gives us $\frac{6}{12}$, which simplifies to $\frac{1}{2}$.

Step 4: Solve for AF:FB

To find the ratio $AF:FB$, we isolate it by dividing both sides of the equation by $\frac{1}{2}$. Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, $1 \div \frac{1}{2} = 1 \cdot 2 = 2$.

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