I think the easiest proof is via geometry; prove triangles ABQ, ABR and APC are all congruent with corresponding sides the medians. Actually, I think you used geometry to correctly assign the x-coordinate of C to be a. However, your analytic geometry proof is also easy:
A classical geometrical proof is sufficient.
[In what follows, ▲ denotes "triangle"; ∟ denotes "angle"].
Consider ▲s ARB, AQB
∟ARB = ∟AQB = 90°
AB is common to both ▲s
Since AC = BC and BR and AQ are medians, AR = BQ.
Thus: ▲ARB & ▲AQB are congruent.
Hence, AQ = BR
Repeat that argument for ▲ARB and ▲BRC.
Result: AQ = BP = BR . . . . as required.
Al. (Skywave) / June 16, 2017