common point of triangle medians
Theorem. The three medians (http://planetmath.org/Median) of a triangle intersect one another in one point, which divides each median in the ratio .
Proof. Let the medians of a triangle be , and . Any median vector is the arithmetic mean of the side vectors emanating from the same vertex. Using vectors, let us form three ways all beginning from the vertex , the first going simply of the median vector ( in the picture):
The second way goes first the side vector and then of the median vector (green in the picture):
Similarly, the third way goes first the side vector and then of the median vector (red in the picture):
Thus the ways (2) and (3), where one goes from to another vertex and continues along the corresponding median of its length, lead to the point which is attained directly along . This means that all medians intersect in . The distance of from any vertex is of the corresponding median, and so the rest of the median is of its length, i.e. the ratio of the parts of any median is .
|Title||common point of triangle medians|
|Date of creation||2013-03-22 17:46:54|
|Last modified on||2013-03-22 17:46:54|
|Last modified by||pahio (2872)|