Triangles And Planets

$ABS$ and $BSC'$ have the same area.

Firstly, $ABS$ and $BSC$ have the same area since they have bases respectively $AB$ and $BC$ of equal length by hypothesis and they have the same height since they share the vertex $S$ .

Secondly $BSC$ and $BSC'$ have the same area since they share the same basis $BS$ and have heigth of same length since by hypothesis $CC'$ is parallel to $BS$ .

Thus by transitivity $ABS$ and $BSC'$ have the same area.

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The interesting bit about this simple problem is that it gives an elementary derivation of Second Kepler's Law by assuming Newton's Gravitational Law : let $S$ be the position of the sun and let $A$ be the starting position of the Earth. After unit time, the new position is $B$ . If there was no force upon the Earth, the earth would move with uniform rectilinear motion and would find itself at place $C$ after anouther unit time.

But since there is a force applied along $BS$ by Newton's Gravitational Law , we apply the Parallelogram Law and we see that the actual position of the earth in another unit of time is $C'$ .

We showed that the $ABS$ and $BSC'$ have the same area, i.e. that in unit of times the vector $BS$ sweeps equal areas, in the limit when the unit time is very small.