To prove that the three medians of a triangle meet at a single point (the centroid), we can use a combination of geometric properties and coordinate geometry.
Geometric Properties:
- A median of a triangle is a line segment connecting a vertex to the midpoint of the opposite side.
- We denote triangle \(ABC\) with vertices \(A\), \(B\), and \(C\), and let \(D\), \(E\), and \(F\) be the midpoints of sides \(BC\), \(CA\), and \(AB\), respectively. Thus, the medians are \(AD\), \(BE\), and \(CF\).
Using Coordinate Geometry: For a concrete proof, let's assign coordinates to the vertices of triangle \(ABC\):
- Let \(A(0, 0)\)
- Let \(B(b, 0)\)
- Let \(C(c_x, c_y)\)
Now, we need to find the midpoints of the sides:
- The midpoint \(D\) of side \(BC\) is given by: \[ D = \left(\frac{b + c_x}{2}, \frac{c_y}{2}\right) \]
- The midpoint \(E\) of side \(CA\) is given by: \[ E = \left(\frac{c_x}{2}, \frac{c_y}{2}\right) \]
- The midpoint \(F\) of side \(AB\) is given by: \[ F = \left(\frac{b}{2}, 0\right) \]
Next, we find the equations of the medians.
Finding the Equation of Median \(AD\):
- The line through points \(A(0, 0)\) and \(D\) has the slope: \[ \text{slope} = \frac{\frac{c_y}{2} - 0}{\frac{b + c_x}{2} - 0} = \frac{c_y}{b + c_x} \]
- The equation of the line can thus be written as: \[ y = \frac{c_y}{b + c_x} x \]
Finding the Equation of Median \(BE\):
- The line through points \(B(b, 0)\) and \(E\) has the slope: \[ \text{slope} = \frac{\frac{c_y}{2} - 0}{\frac{c_x}{2} - b} = \frac{c_y}{c_x - 2b} \]
- The equation of the line can thus be written as: \[ y - 0 = \frac{c_y}{c_x - 2b}(x - b) \] or simplified, it becomes \[ y = \frac{c_y}{c_x - 2b}(x - b) \]
Finding the Intersection Point of the Medians:
- Now we will find the intersection of medians \(AD\) and \(BE\). Setting the equations equal can be complex. However, note that: Since the centroid \(G\) divides each median in a \(2:1\) ratio, we can use this property directly.
- The centroid \(G\) has coordinates: \[ G \left(\frac{0 + b + c_x}{3}, \frac{0 + 0 + c_y}{3}\right) \] which simplifies to: \[ G \left(\frac{b + c_x}{3}, \frac{c_y}{3}\right) \]
Conclusion: By showing that the point \(G\) divides each median in a specific ratio and that our constructed coordinates agree with the geometric definition of a centroid, we conclude that all three medians intersect at the centroid \(G\), a point that is uniquely defined as the average of the triangle's vertices' coordinates.
Thus, the medians of a triangle indeed meet at a single point—the centroid.
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