The mistake is in making differences between positions as they were scalars, while they are vectors.

Consider the vectors $\vec{r}_1$, $\vec{r}_2$, and $\vec{r}_3$ corresponding to the position of $P_1$, $P_2$, and $P_3$ from $A$, respectively. The magnitude of these vectors are the distances of $P_1$, $P_2$, and $P_3$ from $A$. Now, if you just compute the differences between these magnitudes, you get the results in your questions, which are not correct. In fact, you should compute the differences between vectors:

• $\vec{r}_2 - \vec{r}_1$ is the position vector of $P_2$ with respect to $P_1$.
• $\vec{r}_3 - \vec{r}_2$ is the position vector of $P_3$ with respect to $P_2$.

Computing the magnitude of these differences gives you the right result.

Another way to understand the same mistake is the following: take a triangle with side length, $a$, $b$, and $c$. For a generic triangle, $c \neq a + b$. However, in your post you are saying that $c = a + b$, which is indeed incorrect.

The mistake is in making differences between positions as they were scalars, while they are vectors.

Consider the vectors $\vec{r}_1$, $\vec{r}_2$, and $\vec{r}_3$ corresponding to the position of $P_1$, $P_2$, and $P_3$ from $A$, respectively. The magnitude of these vectors are the distances of $P_1$, $P_2$, and $P_3$ from $A$. Now, if you just compute the differences between these magnitudes, you get the results in your questions, which are not correct. In fact, you should compute the differences between vectors:

• $\vec{r}_2 - \vec{r}_1$ is the position vector of $P_2$ with respect to $P_1$.
• $\vec{r}_3 - \vec{r}_2$ is the position vector of $P_3$ with respect to $P_2$.

Computing the magnitude of these differences gives you the right result.

Another way to understand the same mistake is the following: take a triangle with side length $a$, $b$, and $c$. For a generic triangle, $c \neq a + b$. However, in your post you are saying that $c = a + b$, which is indeed incorrect.