# Show, three different points $z_1,z_2,z_3\in\mathbb{C}$ are on a straight line if and only if there is a real number $r$ with $z_3-z_1=r(z_2-z_1).$

If and only if then means that I have to show an equivalence. I could prove one implication, but not the other. The solutions says:

A relation $$z_3-z_1=r(z_2-z_1)$$ with $$r\in\mathbb{R}$$ is equivalent to say the vectors $$z_3-z_1$$ and $$z_2-z_1$$ are $$\mathbb{R}-$$linear dependent

Now I have understood that $$z_3-z_1$$ and $$z_2-z_1$$ are vectors and that they are also linearly dependent. But how does this answer the question? Where is the straight line and what can I say about the points $$z_1,z_2,z_3$$?

Here are my results so far:

Suppose $$z_1,z_2,z_3$$ are on a straight line.

We have to show that the points are $$z_3-z_1$$ and $$z_2-z_1$$ are on the same straight line as the points just mentioned.

The following applies $$z_3=az_1$$

and thus $$z_3-z_1=az_1-z_1=(a-1)z_1$$

So the point $$z_3-z_1$$ also lies on the straight line.

You can argue similarly with the other point. So both points are on the straight line and because of this a $$r\in\mathbb{R}$$ also exists so that $$r(z_2-z_1)=z_3-z_1$$ applies.

I could not prove the other implication

My thoughts so far on this are:

If $$r(z_2-z_1)=z_3-z_1$$ then the points must be $$z_2-z_1$$ and $$z_3-z_1$$ on a straight line. If you add location vectors of points of a straight line you get a location vector for a new point on the same straight line.

So I added the two location vectors and got :

$$z_3-z_1+z_1-z_2=z_3-z_2$$

So the point $$z_3-z_2$$ is on the straight line again.

I then continued to add the location vectors of the 3 points I have so that I can show that at least one of the points $$(z_1,z_2,z_3)$$ can be represented as a sum.

But I didn't make it, I looked into the solutions and expected to find the identity $$r(z_2-z_1)=z_3-z_1$$ I took advantage of him.

My approach must have been a wrong one, I hope someone can explain the solution to me so that I can at least comprehend this implication.

If there is a real number $$r$$ such that $$z_3-z_1=r(z_2-z_1)$$, then consider the straight line$$R=\{z_1+t(z_2-z_1)\,|\,t\in\mathbb{R}\}.$$Then $$z_1,z_2,z_3\in R$$. In fact:
• if you take $$t=0$$, you deduce that $$z_1\in R$$;
• if you take $$t=1$$, you deduce that $$z_2\in R$$;
• if you take $$t=r$$, you deduce that $$z_3\in R$$.