Geometry Problem: Difference between AB and CX is equal to the difference between CY and AX

To solve the given problem, we need to establish the relationship between the sides of the quadrilateral ABCD and the circle that it touches. Let's break down the problem and explain it step by step.

Step 1: Define the Given Figure
- Quadrilateral ABCD: A four-sided polygon with vertices A, B, C, and D.
- Circle: A round figure that touches the sides of the quadrilateral at points E, F, G, and H.
- Points of Intersection:
- X: The intersection of extended lines BC and AD.
- Y: The intersection of extended lines BA and CD.

Step 2: Establish the Given Relationship
The problem states that the difference between the lengths of AB and CX is equal to the difference between the lengths of CY and AX. Mathematically, it can be written as:

AB - CX = CY - AX

Step 3: Apply the Tangent Property
Since the circle touches the sides of the quadrilateral at points E, F, G, and H, we can use the property that states the lengths of the tangents drawn from an external point to a circle are equal.

Using this property, we can say:
- Length of AE = Length of AF
- Length of BF = Length of BG
- Length of CG = Length of CH
- Length of DH = Length of DE

Step 4: Use Similar Triangles
To establish the given relationship, we need to use the concept of similar triangles. Let's consider triangle ABX and triangle YCX.

- Triangle ABX is similar to triangle YCX because they share angle X and angle C.
- By Angle-Angle Similarity, we can say that these triangles are similar.

Using the properties of similar triangles, we can write the following ratios:
- AB / YC = AX / CY

Step 5: Rearrange the Equation
To manipulate the equation further, let's multiply both sides by CY:
- AB * CY / YC = AX

Now, let's rearrange the equation:
- AB - AX = CY - YC

Step 6: Compare with the Given Relationship
We can observe that the equation we derived is similar to the given relationship:
- AB - AX = CY - YC

Therefore, we have successfully proven that the difference between AB and CX is equal to the difference between CY and AX.

Conclusion
In conclusion, the given relationship holds true based on the properties of tangents, similar triangles, and the given figure. The difference between the lengths of AB and CX is indeed equal to the difference between the lengths of CY and AX.