Legendre and the Discovery Section 7 with recording.pptx -...

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Legendre and the Discovery Chapter 2
Legendre’s approach - Also a reductio ad absurdum argument Assume that ΔABC has an angle sum less than two right angles. Let α be the defect. Construct the diagram as mentioned in the text. m 1 =m 2 and AB=CD. ΔABC Δ DCB so has the same angle sum. Extend lines AB and AC so that a line through D intersect both. Call the points of intersection E and F. Thus the angle sum of at most 720°-2α. The angles at C are 180° as are those at D and B. So Δ AEF is at most 180°-2α. Therefore we now have a method for doubling the defect of a triangle. What happens if this continues? The defect can increase without bound, his contradiction since it must be less than 180°. Nothing is wrong with the proof but Legendre used an assumption of the substitution “Through any point within any angle a line can be drawn which meets both side of the angle.

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