Show that in a triangle, the medians are concurrent.
Proof:
Let \(ABC\) be the triangle.
Medians are line segments joining each vertex to the midpoint of the corresponding opposite sides.
Thus, medians are the cevians where \(D\), \(E\), \(F\) are midpoints of \(BC\), \(CA\) and \(AB\), respectively.
Since \(D\) is a midpoint of
\(BD =\)
\(\frac{BD}{DC} = 1\) - - - - (1)
Since \(E\) is a midpoint of
\(CE =\)
\(\frac{CE}{EA} = 1\) - - - - (2)
Since is a midpoint of \(AB\)
\(AF = \)
\(\frac{AF}{FB} = 1\)
Multiplying (1), (2) and (3), we get:
\(\frac{BD}{DC} \times \frac{CE}{EA} \times \frac{AF}{FB} = \)
And so, Ceva's theorem is satisfied,
Hence, the median is concurrent.