In the triangle ABC, let G be the centroid, and let I be the center of the inscribed circle. Let a and b be the angles at the vertices A and B, respectively. Suppose that the segment IG is parallel to AB and that b = 2 tan^-1 (1/3). Find a. 

Given the triangle \( \triangle ABC \) with centroid \( G \) and incenter \( I \), where \( IG \parallel AB \) and \( \angle B = b = 2 \tan^{-1} \left( \frac{1}{3} \right) \), we are to find the measure of angle \( a \) at vertex \( A \).

Let's use the given information systematically:

1. **Centroid \( G \)**: The centroid \( G \) of a triangle is the point of intersection of its medians, which divides each median in the ratio 2:1.

2. **Incenter \( I \)**: The incenter \( I \) is the point of intersection of the angle bisectors of the triangle and is equidistant from all sides of the triangle.

3. **Parallel segments \( IG \parallel AB \)**: Given \( IG \parallel AB \), this implies that \( IG \) and \( AB \) are in the same direction.

Given that \( \angle B = b = 2 \tan^{-1} \left( \frac{1}{3} \right) \), we calculate the tangent of \( b \):
\[
b = 2 \tan^{-1} \left( \frac{1}{3} \right)
\]
Using the tangent double-angle formula:
\[
\tan (2x) = \frac{2 \tan x}{1 - \tan^2 x}
\]
Let \( x = \tan^{-1} \left( \frac{1}{3} \right) \). Then,
\[
\tan x = \frac{1}{3}
\]
\[
\tan (2x) = \frac{2 \cdot \frac{1}{3}}{1 - \left( \frac{1}{3} \right)^2} = \frac{\frac{2}{3}}{1 - \frac{1}{9}} = \frac{\frac{2}{3}}{\frac{8}{9}} = \frac{2}{3} \cdot \frac{9}{8} = \frac{3}{4}
\]
Therefore,
\[
\tan (2x) = \frac{3}{4}
\]
Thus, 
\[
\tan \left( 2 \tan^{-1} \left( \frac{1}{3} \right) \right) = \frac{3}{4}
\]
This implies:
\[
\tan \left( \angle B \right) = \frac{3}{4}
\]

Next, we need to find \( \angle A \). Knowing that \( IG \parallel AB \) and the properties of the centroid and incenter, we can use the angle sum property of the triangle:
\[
\angle A + \angle B + \angle C = 180^\circ
\]
Let \( \angle A = a \) and \( \angle C = c \).

Because \( IG \parallel AB \), the angle opposite to \( \angle B \) at \( G \) is congruent to \( \angle B \), giving another relationship involving \( \angle A \) and \( \angle B \):
\[
a = b
\]

Since \( b = 2 \tan^{-1} \left( \frac{1}{3} \right) \),
\[
a = 2 \tan^{-1} \left( \frac{1}{3} \right)
\]
Thus, the measure of angle \( a \) is:
\[
\boxed{2 \tan^{-1} \left( \frac{1}{3} \right)}
\]

This solution explains how to find \( a \) using properties of the centroid, incenter, and angle calculations in triangles.

Do you want details or have any questions?

1. What is the centroid of a triangle?
2. How do you calculate the tangent of double angles?
3. What are the properties of the incenter in a triangle?
4. How do parallel segments help in determining angles?
5. How do you verify angle calculations in geometric problems?

Tip: Remember that geometric properties and angle relationships are key tools in solving triangle problems.

Explore a triangle geometry problem involving the centroid, incenter, and angle calculations. Discover how to find angle A given the properties of the centroid and incenter, with specific calculations using the tangent of double angles. Suitable for advanced high school students and those interested in geometric problem-solving.

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