Short Answer 
Ratios in trapezoids can be understood by recognizing the similarity of triangles formed by the intersecting diagonals at point P. The parallel sides create proportional relationships between segments, allowing for established ratios (e.g., AX:XB = CY:YD) that exhibit symmetry when comparing segments from different bases.
Step 1: Understand the Structure of the Trapezoid
A trapezoid, such as ABCD, has two parallel sides and diagonals that intersect at point P. This intersection creates four triangles: ABP, BCP, CDP, and DAP. The key to understanding ratios comes from recognizing the similarities between these triangles due to their parallel bases.
Step 2: Establish Similarity of Triangles
The triangles ABP and CDP are similar because they share a common angle at point P and have parallel bases (AB and CD). This means that the corresponding segments of these triangles are proportional. Specifically, the base segments AX and XB are in proportion to CY and YD. The similarity leads to the ratios:
- AX:XB = CY:YD
Step 3: Analyze the Ratios from the Same Side
When comparing the lengths of the segments from the same side of the trapezoid, the ratios shift depending on the base you are examining. For instance, if you look at segment lengths from the base AB, they are in the ratio m:n (AX:XB), while from base CD, the ratio flips to n:m (DY:YC). This consistent relationship highlights the elegant symmetry in trapezoidal geometry.