# An Archimedean Lemma

**Geometry**Level 4

Select two points \(A({x}_{1}, {y}_{1})\), \(B({x}_{2}, {y}_{2})\) on the curve \[y=px^2+qx+r.\] For which point on the curve \(C({x}_{3}, {y}_{3})\), between points A and B, is the area of triangle ABC maximum? Express

\[\frac{{x}_{3}}{{x}_{1}+{x}_{2}}\] as a decimal.