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Please solve #5. 4. Reread Legendre's attempted proof of the parallel postulate in Chap- ter 1. Find the flaw and justify all the steps that
Please solve #5.
4. Reread Legendre's attempted proof of the parallel postulate in Chap- ter 1. Find the flaw and justify all the steps that are correct. Prove the flawed statement in Euclidean geometry. 5. Find the unjustified assumption in the following proof of the par- allel postulate by Farkas Bolyai (see Figure 5.10). Given P not on line 1, PO perpendicular to lato, and line m perpendicular to PO at P. Let n be any line through P distinct from m and Po. We must show that n meets I. Let A be any point between P and Q. Let B be the unique point such that A.Q. B and AQ QB. Let R be the P R Q B Figure 5.10 Attempted proof by Farkas Bolyai. foot of the perpendicular from A to n. Let C be the unique point such that A.R. Cand AR RC. Then A, B, and C are not collinear (else R =P); hence there is a unique circle y passing through them. Since I is the perpendicular bisector of chord AB of y and n is the perpendicular bisector of chord AC of y, I and n meet at the center of y(Exercise 17(b). Chapter 4). 4. Reread Legendre's attempted proof of the parallel postulate in Chap- ter 1. Find the flaw and justify all the steps that are correct. Prove the flawed statement in Euclidean geometry. 5. Find the unjustified assumption in the following proof of the par- allel postulate by Farkas Bolyai (see Figure 5.10). Given P not on line 1, PO perpendicular to lato, and line m perpendicular to PO at P. Let n be any line through P distinct from m and Po. We must show that n meets I. Let A be any point between P and Q. Let B be the unique point such that A.Q. B and AQ QB. Let R be the P R Q B Figure 5.10 Attempted proof by Farkas Bolyai. foot of the perpendicular from A to n. Let C be the unique point such that A.R. Cand AR RC. Then A, B, and C are not collinear (else R =P); hence there is a unique circle y passing through them. Since I is the perpendicular bisector of chord AB of y and n is the perpendicular bisector of chord AC of y, I and n meet at the center of y(Exercise 17(b). Chapter 4)Step by Step Solution
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