There are an infinite number of points in a line. Especially since the definition of a line includes that it is infinitely long. A line *segment* is a piece of that line that has a finite length (defined by two points)... and I think there are an infinite number of points in a segment.

No triangle in the definition of a point. A point is defined at zero volume, area, or length, therefore a triangle would not be able to fit in it.

drmatt is right ... consider it this way: Assume you have a line that is 1 meter long, with end-points at either end. That line also has an infinite number of points in it, since the closer you look at the line, the more points you discover. Since there is no limit to your theoretical ability to zoom in on the line, the number of points discovered is infinite. This is the same paradox that occurs when you are asked to give the length of the shoreline of the island that consists of England, Wales and Scotland. The answer is that the length of the shoreline is infinite. As you zoom in to measure each turn in the shoreline down to rock, grain of sand, atom and sub-atomic particles, the measurement gets longer and longer so that the correct answer to the shoreline question is also infinite. This phenomenon is also evident when studying fractals, as in the Mandelbrot set. The external edge of a Mandelbrot set is also infinite in length, since at every level of detail, the fractal continue to express its outline with an ever-increasing level of complexity.

And a triangle can only exist in 2-dimensional space. A line by definition is one-dimensional. Here's the proof of that: for a triangle to exist within a line, the angles of each corner would have to be 0 degrees, 0 degrees and 180 degrees. -- In reality, a triangle so defined is describing a line.

Curiously, if you consider such a folded-down triangle, while describing a line to nonetheless still be a triangle, them I suppose you could say that the line could contain the triangle. Alternatively, if you define space as having more than three dimensions, with the extra dimensions wrapped around each point, then a triangle that extrudes to 3 extra dimensions could exist within a point (think of a sphere being a close-up of a single point, with a triangle painted on it).

Finally, if you take things all the way to String theory, there is no line, and no triangle, only vibrating lines of energy expressing themselves so as to manifest each of the critical sub-atomic fundamental particles of matter.

So maybe the correct answer is: Yes, and No, ... it depends on how you look at the problem!

The number of points in a line will be equal to the cardinality of the continuum and is the same size as the size of the set of real numbers.

as many as you want... a line its just a representation of a serie of points drawed very near one from the other. A point doesn't have a form because it is the smaller unit in a drawing. If you can define it as a triangle, then it is a figure very small and not a point.

The number of points in a line is infinity.

There are no triangles in points. Points are infinitely small, and triangles are made up of points.