- 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
2 and AB=CD.
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
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
“Through any point within any angle a line can be drawn which meets both side of the