To solve the problem, we need to find the square of the distance between the centroid (G) and the incenter (I) of triangle PQR, where the vertices are given as P(-26, 17), Q(30, 17), and R(10, 2).
### Step 1: Find the Centroid (G) of Triangle PQR
The formula for the centroid \( G \) of a triangle with vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is given by:
\[
G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
\]
Substituting the coordinates of points P, Q, and R:
- \( P(-26, 17) \)
- \( Q(30, 17) \)
- \( R(10, 2) \)
Calculating \( G \):
\[
G_x = \frac{-26 + 30 + 10}{3} = \frac{14}{3}
\]
\[
G_y = \frac{17 + 17 + 2}{3} = \frac{36}{3} = 12
\]
Thus, the centroid \( G \) is:
\[
G = \left( \frac{14}{3}, 12 \right)
\]
### Step 2: Find the Incenter (I) of Triangle PQR
To find the incenter \( I \), we need the lengths of the sides of the triangle:
- \( a = QR \)
- \( b = PR \)
- \( c = PQ \)
Using the distance formula:
\[
QR = \sqrt{(30 - 10)^2 + (17 - 2)^2} = \sqrt{20^2 + 15^2} = \sqrt{400 + 225} = \sqrt{625} = 25
\]
\[
PR = \sqrt{(-26 - 10)^2 + (17 - 2)^2} = \sqrt{(-36)^2 + 15^2} = \sqrt{1296 + 225} = \sqrt{1521} = 39
\]
\[
PQ = \sqrt{(-26 - 30)^2 + (17 - 17)^2} = \sqrt{(-56)^2 + 0^2} = \sqrt{3136} = 56
\]
Now, we can find the incenter \( I \) using the formula:
\[
I_x = \frac{a \cdot x_1 + b \cdot x_2 + c \cdot x_3}{a + b + c}
\]
\[
I_y = \frac{a \cdot y_1 + b \cdot y_2 + c \cdot y_3}{a + b + c}
\]
Substituting the values:
\[
I_x = \frac{25 \cdot (-26) + 39 \cdot 30 + 56 \cdot 10}{25 + 39 + 56}
\]
Calculating the numerator:
\[
= \frac{-650 + 1170 + 560}{120} = \frac{1080}{120} = 9
\]
Now for \( I_y \):
\[
I_y = \frac{25 \cdot 17 + 39 \cdot 17 + 56 \cdot 2}{25 + 39 + 56}
\]
Calculating the numerator:
\[
= \frac{425 + 663 + 112}{120} = \frac{1200}{120} = 10
\]
Thus, the incenter \( I \) is:
\[
I = (9, 10)
\]
### Step 3: Calculate \( GI^2 \)
Now, we can calculate the square of the distance \( GI \):
\[
GI^2 = (G_x - I_x)^2 + (G_y - I_y)^2
\]
Substituting the values:
\[
GI^2 = \left( \frac{14}{3} - 9 \right)^2 + (12 - 10)^2
\]
Calculating \( G_x - I_x \):
\[
= \frac{14}{3} - \frac{27}{3} = \frac{-13}{3}
\]
Calculating \( GI^2 \):
\[
GI^2 = \left( \frac{-13}{3} \right)^2 + (2)^2 = \frac{169}{9} + 4 = \frac{169}{9} + \frac{36}{9} = \frac{205}{9}
\]
### Final Answer
Thus, the value of \( GI^2 \) is:
\[
\boxed{\frac{205}{9}}
\]
