I found in internet the figure of the triangles for this question and attached it to this answer.
Answer: there are many proportions that proves that. One is this: HY / ZY = HI / JI. Below you have the explanation.
Explanation.
SSS stands for Side Side Side.
SSS similarity theorem states that when the corresponding sides of two triangles have the same ratio then the two triangles are similar.
As per the figure the two triangles to compare are HYZ and HIJ.
So, to be similar they must meet:
1) HY / HZ = HI / HJ
2) HY / ZY = HI / JI
Let's probe 2:
From the data shown:
HY = 3,
HZ = 7,
HI = HY + YI = 3 + 3 = 6
ZY = 6
JI = 12
Then,
HY / ZY = 3 / 6 = 1 / 2
HI / JI = 6 / 12 = 1 / 2
So, the proportion HY / ZY = HI / JI proves that triangle similarity.
With this procedure you can find other proportions.
