# Symbolic Identity Report

The following expressions were simplified symbolically.

- **gplm_product**: `0`
- **geometric_balance**: `0`
- **straight_segment_squared_average**: `u**2 + v**2`

Interpretation:

- `gplm_product = 0` proves the two endpoint ratios have fixed product `b/a`.
- `geometric_balance = 0` proves the geometric center equalizes the two ratios.
- the straight-segment expression reduces to `u**2 + v**2`, which is nonnegative.