Step 1: Read the geometric meaning.
For any point P, the triangle inequality gives \(PQ + PR \ge QR\), with equality holding only when P lies on the segment QR itself. So P must be the point where the line \(2x-y-1=0\) crosses segment QR.
Step 2: Write the line QR.
With \(Q(3,4)\) and \(R(1,2)\), the slope of QR is \(\frac{4-2}{3-1}=1\), so QR is \(y - 2 = 1\cdot(x-1)\), which gives \(y = x+1\).
Step 3: Solve the two lines together.
\[2x - y - 1 = 0, \qquad y = x+1\]
Substituting: \(2x - (x+1) - 1 = 0 \implies x - 2 = 0 \implies x = 2\), and then \(y = 2+1 = 3\).
Step 4: Check it lies between Q and R.
\((2,3)\) has x-coordinate between 1 and 3, so it genuinely sits on segment QR, confirming the equality condition holds there.
\[\boxed{(2,3)}\]