If any two sides of a triangle are divided by a line in the same ratio, then the line must be __________ to the third side of the triangle.

Introduction / Context:
This conceptual geometry question is based on a very important theorem about triangles that explains when a line segment inside a triangle is parallel to one of its sides. Understanding this theorem helps in solving many problems involving similar triangles, proportional segments, and midpoints.

Given Data / Assumptions:

• We are given a triangle.
• A line cuts any two sides of the triangle.
• The line divides these two sides in the same ratio.
• We are asked what relation this line has with the third side of the triangle.
• Standard Euclidean geometry is assumed.

Concept / Approach:
The relevant result is the Basic Proportionality Theorem, also called Thales theorem. It states:
If a line divides any two sides of a triangle in the same ratio, then that line is parallel to the third side of the triangle.
Conversely, if a line is drawn parallel to one side of a triangle, it will cut the other two sides in the same ratio. This strong two way relationship links proportional division and parallelism.

Step-by-Step Solution:
Step 1: Consider triangle ABC with points D on AB and E on AC. Step 2: Suppose line DE intersects sides AB and AC such that AD / DB = AE / EC. Step 3: The condition that the ratios are equal means the two sides are divided proportionally. Step 4: By the Basic Proportionality Theorem, if a line divides two sides of a triangle proportionally, the line must be parallel to the third side. Step 5: Therefore DE must be parallel to BC, the third side of the triangle.

Verification / Alternative check:
You can imagine drawing a line parallel to the third side first. Because of similarity of the smaller triangle and the original triangle, ratios of corresponding segments on the other two sides are equal. This confirms that proportional division and parallelism always go together in this setting.

Why Other Options Are Wrong:
Equal: A line cannot be equal to a side of a triangle; equality is not the correct geometric relation here.
Perpendicular: Perpendicular lines form right angles, but that has nothing to do with equal ratios on two sides.
Non parallel: This directly contradicts the theorem; with equal ratios, the line must be parallel, not non parallel.
Collinear: Collinear refers to points on the same straight line, not the relationship between a line inside the triangle and a side of the triangle.

Common Pitfalls:
Students sometimes confuse this theorem with properties of medians or angle bisectors. Others think any line that cuts two sides might be parallel, without checking the proportionality condition. Always remember: equal ratios on two sides imply the line is parallel to the third side, and parallelism in turn guarantees proportional division.

Final Answer:
The line must be parallel to the third side of the triangle.