To prove that \(\frac{GJ}{JI} = \frac{HK}{KI}\) given that line segment JK is parallel to line segment GH, and points J and K are on sides GI and HI of triangle GHI, we can apply the Basic Proportionality Theorem (also known as Thales' theorem).
First, consider triangle GHI with the parallel line segment JK that intersects GI at point J and HI at point K. Since JK is parallel to GH, we can conclude that triangle GHI is divided proportionately by the transversal lines GI and HI.
By the Basic Proportionality Theorem, when a line is drawn parallel to one side of a triangle (in this case, GH) that intersects the other two sides (GI and HI), it divides those two sides proportionally. Thus, it follows that the ratios of the segments created are equal. This means that \(\frac{GJ}{JI} = \frac{HK}{KI}\).
In conclusion, since JK is parallel to GH and intersects the sides of triangle GHI such that it creates segments GJ, JI, HK, and KI, the proportionality of these segments holds true. Therefore, we have shown that \(\frac{GJ}{JI} = \frac{HK}{KI}\), completing the proof.