Given a line $r$ and two points $A$, $B$, what are the conditions for $Area(A,B,C) = constant$ for C in the line?

Question

Besides those trivial cases where the area is zero, of course.

I was doing an exercise with a colleague which had the following points:

$P=(1,0,1)$, $Q=(0,1,1)$, $A=(3,0,2)$, $B=(2,1,2)$

and we should find the point $C$ in the line defined by $P$ and $Q$ which defined a triangle with area equals to $1/2$.

It turns out that this point doesn't exist (the exercise is probably wrong), but I was confused at first. I only believed after plotting in GeoGebra.

Could anyone give me some light?

Answer 1

Just that the line $r$ be parallel to the segment $AB$. In that case you can see that the cross product of the vector $AC$ with the vector $AB$ is constant, and so is the area of the triangle $ABC$, which is half of the cross product, and thus half of the product of the length of $AB$ times the distance of the line from that segment (i.e. from $A$ or from $B$).

Answer 2

I think you misunderstood the problem.

• What conditions have to be posed on $A,B$ and $r$, in order to have the property that for all $C$ in $r$, the area of $ABC$ triangle remains the same?

The answer is in the comments and also the given example with $r=PQ$ illustrates it.