https://file.aiquickdraw.com/mathbot/file/17266644713299mukasqc.jpg

The problem in the image presents a geometric figure of triangle \( \triangle ABC \), with several specific points and conditions provided. Here's a breakdown of the key points:

1. \( G \) is the centroid of triangle \( ABC \), and \( D \) is the midpoint of side \( BC \).
2. A circle is drawn with \( G \) as the center and \( GD \) as the radius.
3. Two tangents from point \( A \) touch the circle at points \( E \) and \( F \), creating line segments \( AE \) and \( AF \).
4. The angles at vertex \( B \) and vertex \( C \) are given as \( \angle B = 40^\circ \) and \( \angle C = 45^\circ \), respectively.
5. We are asked to find the sum \( \angle 1 + \angle 2 \), where \( \angle 1 \) and \( \angle 2 \) are the angles formed by the tangents from \( A \) to the circle (as indicated in the diagram).

### Solution Approach:
- First, note that \( G \), being the centroid, divides the medians of the triangle in a 2:1 ratio, and the circle drawn with \( G \) as the center implies some symmetry.
- Since the two tangents from point \( A \) meet the circle at points \( E \) and \( F \), the property of tangents to a circle (that they are equal in length and the angles subtended by them are equal) plays a role.
- The angles \( \angle 1 \) and \( \angle 2 \) are equal because of the symmetry of the tangents \( AE \) and \( AF \).

Using the angle sum property in triangle \( ABC \):
\[
\angle A + \angle B + \angle C = 180^\circ
\]
\[
\angle A = 180^\circ - (40^\circ + 45^\circ) = 95^\circ
\]
Thus, \( \angle A = 95^\circ \).

Since \( \angle A = \angle 1 + \angle 2 \) (due to the symmetry of tangents from point \( A \) to the circle), the sum \( \angle 1 + \angle 2 \) is:
\[
\boxed{95^\circ}
\]

Would you like more details or have any questions about the steps? Here are some related questions:

1. How do we use the properties of tangents to solve such problems?
2. What is the centroid, and how does it divide the medians of a triangle?
3. How does the angle sum property of triangles help in geometry problems?
4. How can symmetry in geometric figures simplify solving for unknown angles?
5. What is the significance of the incenter and circumcenter in triangle geometry?

**Tip:** Always look for symmetrical properties when working with tangents and circles in geometry.

如右圖，已知 G 點為△ABC 的重心，D 點為 BC 的中點，以 G 點為圓心，GD 為半徑畫一圓，且作 A 點到此圓的兩切線段 AE 與 AF，其中 E、F 為切點。若 ∠B = 40°，∠C = 45°，則 ∠1 + ∠2 = ？

Solve a geometric problem involving a triangle's centroid, tangents to a circle, and angle properties. Given a triangle with ∠B = 40° and ∠C = 45°, find the sum of angles formed by tangents from point A to a circle. Suitable for high school geometry students.

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