Analytisk geometri- problem med korsande räta linjer?

The task is:

In a coordinate system, a square has the corners in points O = (0, 0), A = (1, 0), B = (1, 1) and C = (0, 1). The midpoint of BC is M. The point P on the line OM is connected with A. The midpoint of the section AP is D and the midpoint of the section AB is N. Determine the coordinates of the point P if M, D and N are in a straight line.

I calculated the formulas for two of the straight lines (y = 2x) and (y = -x + 1.5)

However, I do not know how to proceed. There are 3 intersections between the lines, 2 of which are unknown.

I tested writing point D using the distance PA. I used that D in the middle of the distance PA is in both the x-direction and the y-direction in the middle between their coordinates.

(xp − 12, yp − 02) (xp-12, yp-02)

The point P is on the line y = 2x, isf is:

yp = 2xpyp = 2xp

Then the coordinate of D (with a minor unknown) becomes:

(xp − 12, 2xp 2) (xp-12, 2xp 2)

D is the line y = -x + 1.5 Isf:

2xp2 = −xp − 12 + 1.52xp2 = -xp-12 + 1.5

x2 on both sides gives:

2xp = xp − 1 + 32xp = xp-1 + 3

It becomes:

xp = 2xp = 2

Then yp must be:

yp = 2 × 2yp = 2 × 2

yp = 4yp = 4

point P is at (2.4)

But the result says (2/3, 4/3)

Where did I miss / 3? (I came up with a lot myself when I wrote down how I thought here XP)