Question

7. (10) Find the flaw in the following attempted proof of the parallel postulate by Wolfgang Bolyai (Hungarian, 1775 - 1856) (see Fig. 3). Given any point P not on a line l, construct a line 1' parallel to through P in the usual way: drop a perpendicular PQ to / and construct /" perpendicular to PQ. Let I" be any line through P distinct from l'. To see that /" intersects I, pick a point A on PQ between P and Q. Extend PQ beyond Q to a point B so that AQ = QB. Now let R be the foot of the perpendicular from A to l" and extend AR to a point C so that AR- RC. Then A, B, and C are not collinear, so there exists a unique circle passing through them. Since I is the perpendicular bisector of the chord AB of this circle and /" is the perpendicular bisector of the chord AC, and /" must intersect in the center of the circle. Figure 3

Answer 1

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