Find the centroid and incentre of the triangle whose vertices are (1, 2), (2, 3) and (3, 4).

Answer

To find the centroid and incenter of the triangle with vertices at (1, 2), (2, 3), and (3, 4), we will follow these steps:

### Step 1: Find the Centroid
The formula for the centroid (G) of a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3) is given by:

\[ G\left(x, y\right) = \left(\frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}\right) \]

Substituting the given points:
- (x1, y1) = (1, 2)
- (x2, y2) = (2, 3)
- (x3, y3) = (3, 4)

Calculating the x-coordinate of the centroid:
\[ x = \frac{1 + 2 + 3}{3} = \frac{6}{3} = 2 \]

Calculating the y-coordinate of the centroid:
\[ y = \frac{2 + 3 + 4}{3} = \frac{9}{3} = 3 \]

Thus, the centroid G is:
\[ G(2, 3) \]

### Step 2: Find the Incenter
The incenter (I) of a triangle is the point where the angle bisectors of the triangle intersect. It is also the center of the circle inscribed within the triangle. The coordinates of the incenter can be calculated using the formula:

\[ I\left(x, y\right) = \left(\frac{a \cdot x1 + b \cdot x2 + c \cdot x3}{a + b + c}, \frac{a \cdot y1 + b \cdot y2 + c \cdot y3}{a + b + c}\right) \]

Where:
- a, b, c are the lengths of the sides opposite to the vertices (x1, y1), (x2, y2), (x3, y3) respectively.

**Step 2.1: Calculate the lengths of the sides**
- Length of side opposite (1, 2):
\[ a = \sqrt{(2 - 3)^2 + (3 - 4)^2} = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2} \]

- Length of side opposite (2, 3):
\[ b = \sqrt{(1 - 3)^2 + (2 - 4)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2} \]

- Length of side opposite (3, 4):
\[ c = \sqrt{(1 - 2)^2 + (2 - 3)^2} = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2} \]

**Step 2.2: Substitute the values into the incenter formula**
Now substituting a, b, c, and the coordinates of the vertices into the incenter formula:

\[ I\left(x, y\right) = \left(\frac{\sqrt{2} \cdot 1 + 2\sqrt{2} \cdot 2 + \sqrt{2} \cdot 3}{\sqrt{2} + 2\sqrt{2} + \sqrt{2}}, \frac{\sqrt{2} \cdot 2 + 2\sqrt{2} \cdot 3 + \sqrt{2} \cdot 4}{\sqrt{2} + 2\sqrt{2} + \sqrt{2}}\right) \]

Calculating the x-coordinate:
\[ x = \frac{\sqrt{2} + 4\sqrt{2} + 3\sqrt{2}}{4\sqrt{2}} = \frac{8\sqrt{2}}{4\sqrt{2}} = 2 \]

Calculating the y-coordinate:
\[ y = \frac{2\sqrt{2} + 6\sqrt{2} + 4\sqrt{2}}{4\sqrt{2}} = \frac{12\sqrt{2}}{4\sqrt{2}} = 3 \]

Thus, the incenter I is:
\[ I(2, 3) \]

### Final Answer:
- Centroid (G): (2, 3)
- Incenter (I): (2, 3)