All of us know that every object with a mass has a weight center.
This is a powerful theorem and it has important consequences even
for the simplest cases. Before we discuss them, we start with the
straightforward example of a rod with a weight 0 and weights w1
and w2 fastened to the ends of this rod.
Then the weight center of such object is a point P, which is a
point of the interval AB and
AP*w1=BP*w2.
This means that if w1
is bigger and bigger, P is closer and closer to the point A. Let
us consider a straight line L passing through P and perpendicular
to AB. Then AP*w1 and BP*w2 are the weight moments of masses w1
and w2 with respect to L (Since AP is the distance of A to L and
BP is the distance of B to L). This is a general idea of a
weight center - if a straight line L passes through the weight
center, it divides an object into two parts with equal weight
moments with respect to L.
Let us consider a little more advanced problem. Let us consider a
triangle ABC and let us place weights w1,w2, w3 respectively at
the points A,B,C.
Let P be the weight center of the obtained object. Assume that the
straight lines AP, BP, CP cross sides of ABC at the points D, E, F.
Denote by L the straight line BE and by x - angle between BE and AE. Of course the weight moment of w2 (placed at B) with respect to L is 0.
Weight moment of w1 with respect to L is w1*AE*sin(x), weight moment of w3 (placed at C) with respect to L is w3*CE*sin(x). Comparing these two values we see that w1*AE*sin(x)=w3*CE*sin(x).
Doing the same with all vertices we obtain 3 equalities:
which gives us:
If this equality holds, then we can reverse the previous procedure finding w1, w2, w3. We proved the following
Theorem. Let D be a point of side BC of a triangle ABC, E - point of side AC, F - point of side AB. Let L1 be a straight line passing through A and D, L2 - straight line passing through B and E, L3- straight line passing through C and F.
Then L1, L2 and L3 intersect in one point only if and only if
As a consequence of this theorem we conclude that medians of triangle intersect in one point and that bisects of triangle intersect in one point (why?).
Theorem is little known, although it deserves more attention. For 'Literka' it is important that to prove it, physics ideas were used.
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