The image depicts a geometric problem. Based on the annotations and the Chinese text provided, the problem includes:

1. $O$ lies on the segment $AB$.
2. $CO \perp AB$.
3. $AO + BO = CO$.
4. $BO = BE$, $AO = AD$.
5. $\angle ACB = 58^\circ$.

The task is to calculate $\angle DCE$.

Let me analyze and calculate step-by-step. sympy as sp

Define variables and known angles

angle_ACB = 58 # degrees

Use known symmetry and properties to solve for unknowns

The setup is geometric, so most of the work will be symbolic relationships and verification

We know AO = AD, BO = BE, and CO is perpendicular to AB

Define unknown angle DCE symbolically

angle_DCE = sp.Symbol('angle_DCE', positive=True)

Using triangle and angle sum properties to derive relationships

Let's solve to find relationships between angles

angle_sum_triangle = sp.Eq(angle_ACB + angle_DCE + 90, 180) # Triangle sum property including perpendicular

Solve for angle_DCE

angle_DCE_solution = sp.solve(angle_sum_triangle, angle_DCE) angle_DCE_solutionThe calculated value of $\angle DCE$ is $32^\circ$.

This solution relies on the geometric properties and given symmetry in the problem. Let me know if you want a detailed explanation or further assistance.

Here are five questions to expand your understanding:

1. How do perpendicular lines influence angle calculations in triangles?
2. What role does symmetry ($AO = AD$, $BO = BE$) play in solving the problem?
3. How can you confirm that the triangle angle sum property holds in all cases?
4. What are the key steps to determine unknown angles in geometric problems?
5. How does this solution change if one symmetry property is removed?

Tip: Always double-check geometric properties (like perpendicularity and symmetry) for consistency when solving.