Parabolic section

Area(Dark blue triangle) = (1/4)area( light blue triangle)
note: this formula can be applied to any triangle that is drawn in. In general the area of the newer(smaller) triangle will be 1/4th the area of the triangle that was drawn previously.




The formula will be used to find the sum of series in "Quadrature of the parabola" or to approximate the area of a segment of parabola inscribing triangles. Archimedes, using method of exhaustion, constructed more triangles and the area of the triangle, like described above, had 1/4 the area of the triangle drawn in the previous step. The area of the segmet becomes:
                   a +(1/4)a +[(1/4)^2]a+[(1/4)^3]a+...... = k        (any similarities to the geometric series?)

Then k + (1/3)(k-a)=  (4/3)a


Unanswered Questions

1. What were the uses for knowing the area of a segment of paraobla inscribing trianlges  or knowing that a area(one triangle)= (1/4) area(another triangle)?

Evaluation:
Quality: 9
Interesting: 6
complexity: 9