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I would like to know the derivation of the general equation of a conic:

$$Ax^{2} + Bxy + Cy^{2} + Dx + E y + F = 0$$

I have searched over the internet, but I did not find any resource which doesn't make use of trigonometry.

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    What is the equation supposed to govern? Ah nevermind, I missed the title. – 2017-02-28

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To help you get started:

A conic section is the intersection of a cone and a plane.

$$\cases{ax+by+cz +d = 0&\text{plane}\\x^2+y^2 = z^2&\text{cone}}$$

The cone equation comes from the circle equation $x^2+y^2 = R^2$ where we let the third coordinate $z$ steer the radius when increasing.

Then what may be needed is to rotate the result into a plane and maybe let something more be scalable so we get only two coordinates.

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    So ultimately it all comes down to Pythagoras. – 2017-02-28
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    Yes the circle (cone) is Pythagoras. That's where the second order terms come in. – 2017-02-28
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    That guy knew his stuff. – 2017-02-28
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    Yes Pythagoras was quite famous for many things. But maybe most for the right angle triangle formula. – 2017-02-28