Pythagorean Theorem: Basic trigonometry
Pythagorean Theorem:
Some False Proofs
Even smart people make mistakes. Some mistakes are getting published and thus live for posterity to learn from. I 'll list below some fallacious proofs of the Pythagorean theorem that I came across. Some times the errors are subtle and involve circular reasoning or fact misinterpretation. On occasion, a glaring error is committed in logic and leaves one wondering how it could have avoided being noticed by the authors and editors.
Proof 1
One such error appears in the proof X of the collection by B. F. Yanney and J. A. Calderhead (Am Math Monthly, v.3, n. 6/7 (1896), 169-171.)
Suppose the theorem true. Then AB² = AC² + BC², BC² = CD² + BD², and AC² = AD² + CD². Combining the three we get
AB² = AD² + 2CD² + BD².
But CD² = AD·BD. Therefore,
AB² = AD² + 2AD·BD + BD².
From which
AB = AD + BD, which is true. The supposition is true.
Critique
By the same token, assume 1 = 2. Then, by symmetry, 2 = 1. By Euclid 's Second Common Notion, we may add the the two identities side by side: 3 = 3. Which is true, but does not make the assumption(1 = 2) even one bit less false.
As we know, falsity implies anything, truth in particular.
Proof 2
This proof is by E. S. Loomis (Am Math Monthly, v. 8, n. 11 (1901), 233.)
Let ABC be a right triangle whose sides are tangent to the circle O. Since CD = CF, BE = BF, and AE = AD = r = radius of circle, it is easily shown that
(CB = a) + 2r = (AC + AB = b + c).
And if
(1)
a + 2r = b + c then (1)² = (2):
(2)
a² + 4ra + 4r² = b² + 2bc + c².
Now if 4ra + 4r² = 2bc, then a² = b² + c². But 4ra + 4r² is greater than, equal to, or less than 2bc.
If 4ra + 4r² > or < 2bc, then a² + 4ra + 4r² > or < b² + 2bc + c²; i.e. a + 2r < or > b + c, which is absurd. Hence, 4ra + 4r² = 2bc and, therefore, a² = b² + c².
This proof is accompanied by an editors 's Note:
So far as we know, this proof has not been given before. If it has not been published before, it may be properly called a new
Some False Proofs
Even smart people make mistakes. Some mistakes are getting published and thus live for posterity to learn from. I 'll list below some fallacious proofs of the Pythagorean theorem that I came across. Some times the errors are subtle and involve circular reasoning or fact misinterpretation. On occasion, a glaring error is committed in logic and leaves one wondering how it could have avoided being noticed by the authors and editors.
Proof 1
One such error appears in the proof X of the collection by B. F. Yanney and J. A. Calderhead (Am Math Monthly, v.3, n. 6/7 (1896), 169-171.)
Suppose the theorem true. Then AB² = AC² + BC², BC² = CD² + BD², and AC² = AD² + CD². Combining the three we get
AB² = AD² + 2CD² + BD².
But CD² = AD·BD. Therefore,
AB² = AD² + 2AD·BD + BD².
From which
AB = AD + BD, which is true. The supposition is true.
Critique
By the same token, assume 1 = 2. Then, by symmetry, 2 = 1. By Euclid 's Second Common Notion, we may add the the two identities side by side: 3 = 3. Which is true, but does not make the assumption(1 = 2) even one bit less false.
As we know, falsity implies anything, truth in particular.
Proof 2
This proof is by E. S. Loomis (Am Math Monthly, v. 8, n. 11 (1901), 233.)
Let ABC be a right triangle whose sides are tangent to the circle O. Since CD = CF, BE = BF, and AE = AD = r = radius of circle, it is easily shown that
(CB = a) + 2r = (AC + AB = b + c).
And if
(1)
a + 2r = b + c then (1)² = (2):
(2)
a² + 4ra + 4r² = b² + 2bc + c².
Now if 4ra + 4r² = 2bc, then a² = b² + c². But 4ra + 4r² is greater than, equal to, or less than 2bc.
If 4ra + 4r² > or < 2bc, then a² + 4ra + 4r² > or < b² + 2bc + c²; i.e. a + 2r < or > b + c, which is absurd. Hence, 4ra + 4r² = 2bc and, therefore, a² = b² + c².
This proof is accompanied by an editors 's Note:
So far as we know, this proof has not been given before. If it has not been published before, it may be properly called a new
References: 1. E. S. Loomis, The Pythagorean Proposition, NCTM, 1968 |Contact| |Front page| |Contents| |Geometry| |Store| Copyright © 1996-2012 Alexander Bogomolny