I am working on 7.3 in Hartshorne's Euclid and Beyond which states:
Given a triangle $ABC$, show that the sies $\overline{AB}$, $\overline{AC}$, and $\overline{BC}$ and the vertices $A$, $B$, $C$ uniquely determined by the triangle. Hint: Consider the different ways in which a line can intersect the triangle.
I would have thought that a triangle was simply $3$ non-colinear points and that all you need to determine the triangle is, well, $A$, $B$, and $C$ (which are unique by assumption) and that any two points determine a unqiue line segment, but I am unsure of this argument since it doesn't use the hint.
I have been trying to find his definition of "triangle", I also looked at Euclid's elements definitions and could only find 3 definitions about classes of triangles. Also, the idea of angle is covered two sections later as well.
I don't know how to approach this question, because I don't understand what Hartshorne is referring to by a triangle. What definition should I be using?
This question appears similar but is more "real world", and less concerned with axiomatic methods.
Hartshorne defines "triangle" in the Definition on page 74. A triangle is a set of points which is a union of three line segments formed by three noncollinear points. So, a triangle is the union of its sides, considered as a set of points.