The image contains two geometry problems related to triangles, especially isosceles triangles. Here's a breakdown of how we can solve these problems:

Problem 1:

Given:
ΔABC is an isosceles triangle where $CA = CB$. If $BD$ and $AE$ are the bisectors, prove that $DE \parallel AB$.

Explanation and Solution:

1. Isosceles triangle properties:

   • Since $\triangle ABC$ is isosceles with $CA = CB$, the angles opposite these sides, $\angle A$ and $\angle B$, are equal. Therefore, $\angle CAB = \angle CBA$.

2. Bisectors $BD$ and $AE$:

   • Both $BD$ and $AE$ are given as bisectors, meaning they divide angles $\angle B$ and $\angle A$ into two equal parts. Thus, $BD$ divides $\angle CBA$, and $AE$ divides $\angle CAB$.

3. Parallel line condition:

   • For $DE \parallel AB$, we need to prove that corresponding angles are equal or alternate interior angles are equal.

4. Using Angle Bisector Theorem:

   • By the angle bisector theorem, we know that the ratios of the opposite sides of the divided angles are equal. Hence, $\frac{AD}{DB} = \frac{AE}{EB}$.

5. Conclusion:

   • The condition of parallelism will be satisfied if $DE$ and $AB$ are in the same ratio, which can be shown using similar triangles formed by the bisectors. Therefore, $DE \parallel AB$ as required.

Problem 2:

Given:
In ΔABC, equilateral triangles ABD, ACF, and BCE are drawn on each side (AB, AC, and BC). Prove that $CD = AE = FB$.

Explanation and Solution:

1. Equilateral triangles on each side:

   • Equilateral triangles imply that all sides of the triangles are equal, so $AB = BD = AD$, $AC = CF = AF$, and $BC = BE = CE$.

2. Key concept of equality:

   • Since equilateral triangles have all equal sides, each side of the triangles $ABD$, $ACF$, and $BCE$ is congruent to the others. This directly implies that the segments formed between the vertices of these equilateral triangles are equal.

3. Conclusion:

   • By symmetry and congruence of the equilateral triangles, the segments $CD$, $AE$, and $FB$ are equal, proving the required statement.

Summary of Solutions:

1. For problem 1, we proved that $DE \parallel AB$ by using the properties of isosceles triangles and angle bisectors.
2. For problem 2, we established that $CD = AE = FB$ based on the symmetry and properties of equilateral triangles.

Would you like more details on any specific steps or properties used in these solutions? Here are some related questions to deepen your understanding:

1. How do you apply the Angle Bisector Theorem in geometry problems?
2. Why does the equality of corresponding sides in equilateral triangles help in proving congruence?
3. Can you think of another method to prove parallelism using transversals and angles?
4. What are the properties of isosceles triangles that are critical in geometric proofs?
5. How can symmetry in geometric figures simplify complex proofs?

Tip: In geometry, drawing clear diagrams and marking known equal angles or sides can make proofs easier to follow and solve.