Concept guide
Pythagorean theorem: a proof by dot products in the coordinate model
In the Euclidean plane represented by ℝ² (with Euclidean length as distance), prove: for every right triangle ABC with the right angle at C, legs of lengths a=d(C,B) and b=d(C,A), and hypotenuse of length c=d(B,A), a²+b²=c². Then check the example C=(1,2), B=(4,6), A=(−7,8) (25+100=125), and use the non-right triangle E=(0,0), F=(4,0), H=(1,2) to show that without a right angle the relation fails in general.