explanation

The incenter of a triangle is given by:

$$ C = \left( ~ \frac{a \cdot A_x + b \cdot B_x + c \cdot C_x}{a + b + c}~,~ \frac{a \cdot A_y + b \cdot B_y + c \cdot C_y}{a + b + c} \right) $$

where $ a = d(B,C)$ , $ b = d(A,C) $ and $ c = d(A,B) $.

In this example we have : $ A_x = 1 $ , $ A_y = -17 $ , $ B_x = -5 $ , $ B_y = 0 $ , $ C_x = -8 $ and $ C_y = -4 $ .

The sides lengths are: $ a = 5 $ , $ b = 5 \sqrt{ 10 } $ and $ c = 5 \sqrt{ 13 } $.

When we insert these values into the formula, we obtain the given result.

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