Editing Parabola (section)

=== Derivation of quadratic equation ===
The lengths of {{overline|BM}} and {{overline|CM}} are:
{{unbulleted list | style = padding-left:1.6em;
| <math>\overline\mathrm{BM} = 2y\cos\theta</math>{{pad|1em}}(triangle BPM is [[isosceles]], because <math>\overline{PM} \parallel \overline{AC} \implies \angle PMB = \angle ACB = \angle ABC</math>
| <math>\overline\mathrm{CM} = 2r</math>{{pad|1em}}(PMCK is a [[parallelogram]]).
}}
Using the [[Chord theorem|intersecting chords theorem]] on the chords {{overline|BC}} and {{overline|DE}}, we get
<math display="block">\overline\mathrm{BM} \cdot \overline\mathrm{CM} = \overline\mathrm{DM} \cdot \overline\mathrm{EM}.</math>

Substituting:
<math display="block">4ry\cos\theta = x^2.</math>

Rearranging:
<math display="block">y = \frac{x^2}{4r\cos\theta}.</math>

For any given cone and parabola, {{mvar|r}} and {{mvar|θ}} are constants, but {{mvar|x}} and {{mvar|y}} are variables that depend on the arbitrary height at which the horizontal cross-section BECD is made. This last equation shows the relationship between these variables. They can be interpreted as [[Cartesian coordinate system|Cartesian coordinates]] of the points D and E, in a system in the pink plane with P as its origin. Since {{mvar|x}} is squared in the equation, the fact that D and E are on opposite sides of the {{mvar|y}} axis is unimportant. If the horizontal cross-section moves up or down, toward or away from the apex of the cone, D and E move along the parabola, always maintaining the relationship between {{mvar|x}} and {{mvar|y}} shown in the equation. The parabolic curve is therefore the [[Locus (mathematics)|locus]] of points where the equation is satisfied, which makes it a [[Graph of a function|Cartesian graph]] of the quadratic function in the equation.