Given:
- Quadrilateral ABCD
- Diagonal AC and BD intersect at point X
- E, F, G, and H are the midpoints of AX, BX, CX, and DX respectively

To find:
The ratio of (EF, FG, GH, GE) to (AD, DC, CB, BA)

Explanation:
Let's first understand the properties of midpoints in a quadrilateral:

1. Midpoint of a diagonal: The midpoint of a diagonal of a quadrilateral divides the diagonal into two equal parts.

2. Midpoint of a side: The midpoint of a side of a quadrilateral divides the side into two equal parts.

Using these properties, let's find the ratio of the segments:

Segment EF:
E is the midpoint of AX, so AE = EX. Similarly, F is the midpoint of BX, so BF = FX. Therefore, EF represents the segment between AE and BF, which is equal to EX + FX.

Segment FG:
F is the midpoint of BX, so BF = FX. Similarly, G is the midpoint of CX, so CG = GX. Therefore, FG represents the segment between BF and CG, which is equal to FX + GX.

Segment GH:
G is the midpoint of CX, so CG = GX. Similarly, H is the midpoint of DX, so DH = HX. Therefore, GH represents the segment between CG and DH, which is equal to GX + HX.

Segment GE:
G is the midpoint of CX, so CG = GX. Similarly, E is the midpoint of AX, so AE = EX. Therefore, GE represents the segment between CG and AE, which is equal to GX + EX.

Segment AD:
A is one endpoint of the diagonal AC, and D is the other endpoint of the diagonal AC. Therefore, AD represents the entire diagonal AC.

Segment DC:
D is one endpoint of the diagonal BD, and C is the other endpoint of the diagonal BD. Therefore, DC represents the entire diagonal BD.

Segment CB:
C is one endpoint of the diagonal BD, and B is the other endpoint of the diagonal BD. Therefore, CB represents the entire diagonal BD.

Segment BA:
B is one endpoint of the diagonal AC, and A is the other endpoint of the diagonal AC. Therefore, BA represents the entire diagonal AC.

Ratio Calculation:
Now, let's find the ratio of the segments:
(EF + FG + GH + GE) / (AD + DC + CB + BA)
= (EX + FX) + (FX + GX) + (GX + HX) + (GX + EX) / (AC + BD)
= 2(EX + FX + GX + HX) / (AC + BD)
= 2(EX + FX + GX + HX) / 2(AC)
= (EX + FX + GX + HX) / AC

As per the properties of midpoints, EX = FX = GX = HX, and since E, F, G, and H are midpoints, they divide the diagonal AC into four equal parts. Therefore, EX + FX + GX + HX = AC.

So, the ratio simplifies to:
(EX + FX + GX + HX