= Quarter-chord reference of a mean aerodynamic chord
{title2=$x_{1/4}=x_{\rm LE,MAC}+\bar c/4$}
The \b[<quarter-chord reference of a mean aerodynamic chord>] is the geometric point one quarter of the reference-chord length aft of its specified leading-edge position:
$$
x_{1/4}=x_{\rm LE,MAC}+\frac{\bar c}{4}.
$$
It supplies a reproducible location for reporting positions and <pitching moments>. The length of the <mean aerodynamic chord> alone does not locate this point; its reference origin is also required.
For a symmetric <wing planform> using the area-weighted placement in <mean aerodynamic chord reference position>,
$$
\bar c=\frac2S\int_0^{b/2}c(y)^2\,dy,\qquad
x_{\rm LE,MAC}=\frac2S\int_0^{b/2}x_{\rm LE}(y)c(y)\,dy.
$$
Adding one quarter of the first integral to the second gives
$$
x_{1/4}=\frac2S\int_0^{b/2}\left[x_{\rm LE}(y)+\frac{c(y)}4\right]c(y)\,dy.
$$
It is consequently an <area>-weighted average of local quarter-chord coordinates under this convention. No aerodynamic-load distribution was needed for that identity.<Applied Computational Aerodynamics by W. H. Mason>[Appendix A, pp. A-27–A-28, wing-area and reference-chord integrals, equations (A-88)–(A-94)]{ref}
For an original trapezoidal example, let span $b=8$ m, root chord 3 m and tip chord 1 m. Then $S=16\,\mathrm{m^2}$, $\bar c=13/6$ m and the positive half-wing reference station is $y_{\rm MAC}=5/3$ m. With straight leading edge $x_{\rm LE}(y)=0.2\,\mathrm m+0.3y$, its reference leading edge is at 0.7 m. The quarter-chord point is therefore \b[$x_{1/4}=1.241667$ m] aft of the stated origin. Changing sweep changes this coordinate even if all spanwise chord lengths remain fixed.
The <aerodynamic center> is defined through the response of <pitching moment> to incidence, rather than through a geometric averaging rule. In ideal two-dimensional <thin airfoil theory>, the quarter-chord moment is independent of <angle of attack>, which motivates that familiar reference. It does not prove that every finite wing's <aerodynamic center> lies at this geometric point.<F3 Lecture Notes: Thin-Airfoil Analysis Problem, continued by MIT OpenCourseWare>[pp. 3–4, quarter-chord moment and aerodynamic-center conclusion within thin-airfoil analysis]{ref}
Moment comparisons must also use the same reference. Moving the reference by a vector $\Delta\boldsymbol r$ changes the moment by $-\Delta\boldsymbol r\times\boldsymbol F$ for the same resultant force. A location convention thus has a measurable mechanical consequence even though it is chosen geometrically.
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