# How does quadrea relate to area?

In NJ Wildeberger's book (which I don't see much use in citing, since its an offline material, so instead I will provide a link to his key points further down) he argues that "Quadrance" can be just as useful as "Distance". He defines Quadrance as: "A squared plus B squared" & "(((C - A)^2)+((B - D)^2)) given lines 'AB' and 'CD'. More info here: " https://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax " & here: " https://en.m.wikipedia.org/wiki/Rational_trigonometry " . Now knowing quadrance fills the shoes of distance; what fills the shoes of area? And how can it be converted back into its 'non-rational' equivalent?

• You need to give more context. According to wikipedia, this is a relatively obscure topic in mathematics and I think that "quadrea" is not even part of the standard theory of rational trigonometry. Dec 22, 2017 at 10:31
• @Raskolnikov Okay, is it alright if I spend a couple hours thinking about how to reframe it in the clearest possible context? Or should I just delete my question and try all over again? (I'm new) Dec 22, 2017 at 10:45
• I think you can rework the question. That will bump it. Dec 22, 2017 at 10:47
• I have source material I could cite, but I made the mistake of assuming that it was common knowledge Dec 22, 2017 at 10:47
• Alright; I will re-work it, but due to limited char space I'm gonna need to put strategy into it Dec 22, 2017 at 10:48

The definition of quadrea is $(Q_1+Q_2+Q_3)^2-2(Q_1^2+Q_2^2+Q_3^2).\;$ If $\;d_21^2=Q_1,\;$ $d_2^2=Q_2,$ $d_3^2=Q_3$, then it is $(d_1+d_2+d_3)(-d_1+d_2+d_3)(d_1-d_2+d_3)(d_1+d_2-d_3)$ and by Heron's formula this is $16$ times the square of the area of the triangle.
This can be checked by looking at an equilateral triangle side $d$ whose area squared is $\frac3{16}d^4.$