If two transverse lines cross parallel lines, transversal line-segment ratios are equal {intercept theorem}.
Pappus invented line and centroid theorems {Pappus's theorems} {Pappus theorems}.
Draw line segments from first first-line-segment point to first second-line-segment point, second first-line-segment point to second second-line-segment point, and first first-line-segment point to second second-line-segment point. The drawn line segments have three intersection points that make a straight line.
Arcs have planes. Arcs rotated around plane axis generate area equal to arc length times circular-path segment traveled by arc centroid.
Planes rotated around plane axis generate volume equal to surface area times circular-path segment traveled by centroid.
revolution
If centroid paths are circles, the previous two theorems generate surfaces or volumes of revolution.
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Date Modified: 2022.0225