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Intersecting lines are foundational elements in geometry. When two lines cross each other, they meet at a single point. This point is known as the 'point of intersection'. Imagine two paths in the woods that cross; where the paths meet, that's like the point of intersection on the lines.

Now, the special thing about intersecting lines in geometry is that they always lie on the same flat surface, known as a plane. No matter how lengthy the lines or where they go, as long as they intersect, they share this plane. This is akin to two streets intersecting in a flat city park; the park is the plane containing the pathways. Intersecting lines are important because they help us understand the concept of planes and space in geometry.

Picture a huge sheet of paper that extends endlessly in all directions. That's like a plane in geometry: a flat, two-dimensional surface that goes on infinitely. A plane doesn't have thickness or edges, and it's determined uniquely by any set of three non-collinear points.

In a more practical sense, think of a plane as a tabletop surface. Anything you place on it that doesn't share a line can be a representation of points defining the plane. Geometrically, once you've got those three separate points, you can say they 'span' a plane, like pegs in a tent pulling the fabric taut to create a flat area. This concept is critical because it's how we start to make sense of space and understand the nature of two-dimensional areas.

Non-collinear points are like friends standing apart in a park, where no single straight path can connect them all without changing direction. In geometry, points are considered collinear if you can draw a straight line that passes through all of them. If you can't do this, the points are non-collinear.

Using non-collinear points is how we define a plane because if they were all in a row (collinear), they would all lie on a single line, and there would be no 'width' to create a flat surface. But once the points are scattered, not in a straight line, they give us the 'corners' of our geometric 'tent', ensuring there's an area spanned by these points. This concept is essential for constructing models in geometry where we need to define areas and not just lines.