If C is the midpoint of AB and P in any point outside AB, then
Here is the MathJax representation of your given vector equations: \[ \overrightarrow{PA} = \overrightarrow{PC} + \overrightarrow{CA} \] \[ \overrightarrow{PB} = \overrightarrow{PC} + \overrightarrow{CB} \] \[ \overrightarrow{PA} + \overrightarrow{PB} = 2\overrightarrow{PC} + (\overrightarrow{CA} + \overrightarrow{CB}) \] \[ = 2\overrightarrow{PC} \quad (\because \overrightarrow{CA} = -\overrightarrow{CB}) \] \[ \text{Hence } \overrightarrow{CA} + \overrightarrow{CB} = 0 \] \[ \text{and } \overrightarrow{PA} + \overrightarrow{PB} = 2\overrightarrow{PC} \]
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