Answer:
See Below. 
Step-by-step explanation:
We are given the isosceles triangle ΔABC. By the definition of isosceles triangles, this means that ∠ABC = ∠ACB. 
Segments BO and CO bisects ∠ABC and ∠ACB. 
And we want to prove that ΔBOC is an isosceles triangle. 
Since BO and CO are the angle bisectors of ∠ABC and ∠ACB, respectively, it means that ∠ABO = ∠CBO and ∠ACO = ∠BCO. 
And since ∠ABC = ∠ACB, this implies that: 
∠ABO = ∠CBO =∠ACO = ∠BCO. 
This is shown in the figure as each angle having only one tick mark, meaning that they are congruent. 
So, we know that: 

∠ABC is the sum of the angles ∠ABO and ∠CBO. Likewise, ∠ACB is the sum of the angles ∠ACO and ∠BCO. Hence: 

Since ∠ABO =∠ACO, by substitution: 

Subtracting ∠ABO from both sides produces: 

So, we've proven that the two angles are congruent, thereby proving that ΔBOC is indeed an isosceles triangle.