This is a geometry problem involving a triangle $\triangle ABC$ with an incircle, and you are asked to determine $\angle 1 + \angle 2$. Let's break down the given information:

• $G$ is the centroid of $\triangle ABC$.
• $D$ is the midpoint of side $BC$, and line $GD$ is the radius of the circle centered at $G$.
• $A$ is a point on the circle, and $E$ and $F$ are points where the circle touches lines $AE$ and $AF$, respectively.
• Given that:
  • $\angle B = 40^\circ$
  • $\angle C = 45^\circ$

We need to find the sum of $\angle 1$ and $\angle 2$.

Step-by-step solution:

1. Angle Sum of Triangle: Using the sum of angles in triangle $ABC$, we have: $\angle A + \angle B + \angle C = 180^\circ$ Substituting the known values $\angle B = 40^\circ$ and $\angle C = 45^\circ$, we can solve for $\angle A$: $\angle A = 180^\circ - 40^\circ - 45^\circ = 95^\circ$

2. Properties of Tangent Lines: Since $E$ and $F$ are points of tangency, and the tangents from a point to a circle are equal in length, we know that: $\angle 1 = \angle 2$

3. External Angles: The angles $\angle 1$ and $\angle 2$ are external angles formed by the tangents to the circle. These external angles are related to the angle at vertex $A$.

4. Final Calculation: The sum of these angles is related to half of the angle $\angle A$, as follows: $\angle 1 + \angle 2 = \frac{1}{2} \times \angle A = \frac{1}{2} \times 95^\circ = 47.5^\circ$

Thus, the value of $\angle 1 + \angle 2$ is $47.5^\circ$.

Would you like further clarification on any steps or additional details?

Here are 5 related questions to expand on this:

1. What is the centroid of a triangle, and how is it constructed?
2. How do you prove that the tangents from a point to a circle are equal?
3. Can we generalize the relation between external angles and vertex angles for any circumscribed triangle?
4. How does the inscribed circle's properties help solve angle-related problems?
5. What are the properties of the median in a triangle, especially when related to the centroid?

Tip: Always remember that the tangents from an external point to a circle are equal in length, a crucial property when working with circumscribed circles.