Illustration of triangle medians and perpendicular bisectors, highlighting the centroid and circumcenter.

The provided image explains two important concepts in geometry: the Centroid and the Circumcentre of a triangle. Let's break down each concept, solve related problems, and explain the solution step by step.

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1. Centroid

#### Definition:
The Centroid of a triangle is the point where all three medians of the triangle intersect. A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.

#### Properties of the Centroid:
- The Centroid divides each median into a ratio of 2:1, with the longer segment being closer to the vertex.
- The Centroid is always located inside the triangle.

#### Example Problem:
Suppose you are given a triangle \( \triangle ABC \) with vertices \( A(0, 0) \), \( B(4, 0) \), and \( C(2, 6) \). Find the coordinates of the Centroid.

#### Solution:
1. Find the midpoints of the sides:
- Midpoint of \( BC \):
\[
M_{BC} = \left( \frac{4+2}{2}, \frac{0+6}{2} \right) = (3, 3)
\]
- Midpoint of \( AC \):
\[
M_{AC} = \left( \frac{0+2}{2}, \frac{0+6}{2} \right) = (1, 3)
\]
- Midpoint of \( AB \):
\[
M_{AB} = \left( \frac{0+4}{2}, \frac{0+0}{2} \right) = (2, 0)
\]

2. Equation of one median (e.g., from \( A \) to \( M_{BC} \)):
- Slope of \( AM_{BC} \):
\[
\text{slope} = \frac{3-0}{3-0} = 1
\]
- Equation of the line \( AM_{BC} \):
\[
y - 0 = 1(x - 0) \implies y = x
\]

3. Equation of another median (e.g., from \( B \) to \( M_{AC} \)):
- Slope of \( BM_{AC} \):
\[
\text{slope} = \frac{3-0}{1-4} = -1
\]
- Equation of the line \( BM_{AC} \):
\[
y - 0 = -1(x - 4) \implies y = -x + 4
\]

4. Find the intersection of the medians \( y = x \) and \( y = -x + 4 \):
\[
x = -x + 4 \implies 2x = 4 \implies x = 2
\]
\[
y = x = 2
\]

5. Coordinates of the Centroid:
\[
\text{Centroid} = (2, 2)
\]

Thus, the Centroid of \( \triangle ABC \) is:
\[
\boxed{(2, 2)}
\]

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2. Circumcentre

#### Definition:
The Circumcentre of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect. It is the center of the circle that passes through all three vertices of the triangle (the circumcircle).

#### Properties of the Circumcentre:
- The Circumcentre may lie inside, on, or outside the triangle, depending on whether the triangle is acute, right, or obtuse, respectively.
- The Circumcentre is equidistant from all three vertices of the triangle.

#### Example Problem:
Given a triangle \( \triangle DEF \) with vertices \( D(0, 0) \), \( E(6, 0) \), and \( F(3, 6) \), find the coordinates of the Circumcentre.

#### Solution:
1. Find the midpoints of the sides:
- Midpoint of \( EF \):
\[
M_{EF} = \left( \frac{6+3}{2}, \frac{0+6}{2} \right) = \left( \frac{9}{2}, 3 \right)
\]
- Midpoint of \( DF \):
\[
M_{DF} = \left( \frac{0+3}{2}, \frac{0+6}{2} \right) = \left( \frac{3}{2}, 3 \right)
\]
- Midpoint of \( DE \):
\[
M_{DE} = \left( \frac{0+6}{2}, \frac{0+0}{2} \right) = (3, 0)
\]

2. Equation of the perpendicular bisector of \( EF \):
- Slope of \( EF \):
\[
\text{slope} = \frac{6-0}{3-6} = -2
\]
- Slope of the perpendicular bisector:
\[
\text{slope} = \frac{1}{2}
\]
- Equation of the perpendicular bisector:
\[
y - 3 = \frac{1}{2} \left( x - \frac{9}{2} \right) \implies y - 3 = \frac{1}{2}x - \frac{9}{4} \implies 4y - 12 = 2x - 9 \implies 2x - 4y + 3 = 0
\]

3. Equation of the perpendicular bisector of \( DF \):
- Slope of \( DF \):
\[
\text{slope} = \frac{6-0}{3-0} = 2
\]
- Slope of the perpendicular bisector:
\[
\text{slope} = -\frac{1}{2}
\]
- Equation of the perpendicular bisector:
\[
y - 3 = -\frac{1}{2} \left( x - \frac{3}{2} \right) \implies y - 3 = -\frac{1}{2}x + \frac{3}{4} \implies 4y - 12 = -2x + 3 \implies 2x + 4y - 15 = 0
\]

4. Solve the system of equations:
\[
2x - 4y + 3 = 0 \quad \text{(1)}
\]
\[
2x + 4y - 15 = 0 \quad \text{(2)}
\]
Add equations (1) and (2):
\[
(2x - 4y + 3) + (2x + 4y - 15) = 0 \implies 4x - 12 = 0 \implies x = 3
\]
Substitute \( x = 3 \) into equation (1):
\[
2(3) - 4y + 3 = 0 \implies 6 - 4y + 3 = 0 \implies 9 - 4y = 0 \implies y = \frac{9}{4}
\]

5. Coordinates of the Circumcentre:
\[
\text{Circumcentre} = \left( 3, \frac{9}{4} \right)
\]

Thus, the Circumcentre of \( \triangle DEF \) is:
\[
\boxed{\left( 3, \frac{9}{4} \right)}
\]

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Summary:

- The Centroid is the intersection of the medians and divides each median in a 2:1 ratio.
- The Circumcentre is the intersection of the perpendicular bisectors and is the center of the circumcircle.

These points are crucial in solving various geometric problems involving triangles.