It is possible to prove that the Cartesian coordinate plane is a Euclidean plane with practically no axioms. We define a point to be an ordered pair of real numbers, and the plane to be the set of all ordered pairs of real numbers. This enables us to use what we know about real numbers. The axioms are the axioms for the development of the real number system.
An important definition is the distance between two points.
Definition: Let (x1, y1) and (x2, y2) be two points. The distance between them is
What we are doing with this definition is we are actually assuming the Pythagorean Theorem, which is well known to be equivalent to Euclid’s Parallel Postulate. However, we are not assuming the whole Pythagorean Theorem, just the special case where the perpendicular sides of the triangle are horizontal and vertical. It is possible to prove the entire Pythagorean Theorem from this special case.
Let ABC be a right triangle with right angle at C.
If we let A = (x1, y1), B = (x2, y2), and C = (x3, y3), then to say that C is the right angle is to say that the slopes of AC and BC are negative reciprocals of each other.
Note that If either if these denominators is 0, then the lines are horizontal or vertical, and we have dealt with that case with the definition of distance, so we do not have to worry about zero denominators.
(y1 – y3)(y2 – y3) = -(x1 – x3)(x2 – x3)
Transpose to get rid of the negative
(y1 – y3)(y2 – y3) + (x1 – x3)(x2 – x3) = 0
y1y2 – y1y3 – y2y3 + y32 + x1x2 – x1x3 – x2x3 + x32 = 0
Transpose all the terms that do not involve x3 or y3 to the other side of the equation.
– y1y3 – y2y3 + y32 – x1x3 – x2x3 + x32 = - y1y2 - x1x2
Double both sides.
– 2y1y3 – 2y2y3 + 2y32 – 2x1x3 – 2x2x3 + 2x32 = - 2y1y2 - 2x1x2
Add x12 + x22 + y12 + y22 to both sides and split up the 2x32 and 2y32.
x12 – 2x1x3 + x32 + y12 – 2y1y3 + y32 + x22 – 2x2x3 + x32 + y22 - 2y2y3 +y32 = x12 – 2x1x2 + x22 + y12 – 2y1y2 + y22
(x1 – x3)2 + (y1 – y3)2 + (x2 – x3)2 + (y2 – y3)2 = (x1 – x2)2 + (y1 – y2)2
a2 + b2 = c2
This proves that if the triangle is a right triangle, the sides are related by the Pythagorean Theorem. For the converse, simply reverse the steps. This direction is actually more straightforward than the way we did it above. The things that we had to pull out of the air to put into both sides of the equation will automatically cancel out. Where we multiplied the binomials in the fourth step to remove the parentheses, we will need to factor when we reverse our steps, and that may require a bit of cleverness, but the steps are reversible, and we have the implication both ways. If it is a right triangle, then a2 + b2 = c2, and if a2 + b2 = c2, then the triangle is a right triangle.