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.