To visualize the problem, we start by sketching triangle \( \triangle GHI \) on a blank sheet of paper. Here's how you might imagine it:
- Draw a triangle with vertices labeled \( G \), \( H \), and \( I \).
- Label the sides accordingly, so \( GH \) is the side between points \( G \) and \( H \), \( GI \) between points \( G \) and \( I \), and \( HI \) between points \( H \) and \( I \).
- Next, you will need to sketch line segment \( JK \) parallel to side \( GH \).
- Place point \( J \) on side \( GI \) and point \( K \) on side \( HI \).
By the construction of line segment \( JK \):
- Since \( JK \) is parallel to \( GH \), we invoke the Basic Proportionality Theorem (also known as Thales's theorem), which states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
To demonstrate this, consider:
- Let \( GJ \) represent the segment from \( G \) to \( J \) on side \( GI \) and \( HK \) represent the segment from \( H \) to \( K \) on side \( HI \).
- According to the theorem, since \( JK \) is parallel to \( GH \), the ratios of the segments on the sides it intersects are equal. In mathematical terms, we express this as:
\[ \frac{GJ}{JI} = \frac{HK}{KI} \]
This shows that the line segment \( JK \) divides the sides \( GI \) and \( HI \) into two segments each, maintaining a proportional relationship.
In conclusion, the proof reveals that any line drawn parallel to one side of a triangle will divide the other two sides proportionally. Thus, in triangle \( \triangle GHI \), the points \( J \) on \( GI \) and \( K \) on \( HI \) create segments that are proportional because \( JK \) is parallel to \( GH \).
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