The relationship between just the sides is that the sum of any two of them must be greater than the third. Any other relationship involves one (or more) angles.
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There is no relationship between slope and the theorem, however the theorem does deal with the relationship between angles and sides of a triangle.
They are of the same lengths
if any two angles are similar the triangle will be similar
The three vertices of the triangle uniquely determine a circle that circumscribes the triangle. The three sides of the triangle uniquely determine the circle that inscribes the triangle.
An equilateral triangle is a triangle with three equal sides. An isosceles triangle is one with two equal sides. So yes, an equilateral triangle qualifies as being an isosceles triangle as well. This is quite similar to the relationship between squares and rectangles, where a square is always a rectangle, but a rectangle isn't necessarily a square.