The image contains a geometric problem in Vietnamese. I will translate the problem into English and then solve it.

Problem Translation:

Given:

• $\Delta ABC$, $D$ is the midpoint of $AC$.
• $E$ is the midpoint of $BC$.
• $BD$ intersects $AE$ at point $O$.

a) Prove that $\triangle AOB = \triangle COB$.

b) Prove that $\angle AOB + \angle BOC = 180^\circ$.

Solution:

Part a) Prove that $\triangle AOB = \triangle COB$.

Since $D$ and $E$ are midpoints of $AC$ and $BC$ respectively, we have:

1. $AD = DC$ and $BE = EC$.
2. $BD$ and $AE$ are the medians of $\triangle ABC$.
3. $O$ is the centroid of $\triangle ABC$, meaning $AO = \frac{2}{3}AE$ and $BO = \frac{2}{3}BD$.

Now, let's consider the triangles $\triangle AOB$ and $\triangle COB$:

• $AB$ is common to both triangles.
• $\angle AOB = \angle COB$ since these angles are vertically opposite.
• $BO = BO$ (common side).

Therefore, by the Side-Angle-Side (SAS) criterion, $\triangle AOB = \triangle COB$.

Part b) Prove that $\angle AOB + \angle BOC = 180^\circ$.

From the result of part a), we know that $\triangle AOB$ and $\triangle COB$ are congruent.

Since $\angle AOB$ and $\angle BOC$ are vertically opposite angles, they must be equal. Let $\angle AOB = \theta$.

Thus, $\angle AOB + \angle BOC = \theta + \theta = 2\theta$.

However, since these angles are on a straight line formed by $AO$ and $BO$, we have:

$\angle AOB + \angle BOC = 180^\circ$

Hence, we have proven that $\angle AOB + \angle BOC = 180^\circ$.

Do you want details or have any questions? Here are 5 related questions:

1. How can we determine the centroid of a triangle given the coordinates of its vertices?
2. What are the properties of medians in a triangle?
3. Can you explain the Side-Angle-Side (SAS) criterion in detail?
4. How does the centroid of a triangle divide the medians?
5. What other methods can be used to prove angle relationships in geometric figures?

Tip: When working with triangle properties, remember that the centroid divides each median into a 2:1 ratio, which is crucial in many geometric proofs.