This is what Loomis claimed in his 1907 book
A New Approach to Proving the Pythagorean Theorem
C-K Shene
Current Version: February 28, 2024
Abstract
Ne’Kiya D. Jackson and Calcea Rujean Johnson presented a trigonometric proof of the
Pythagorean Theorem at the 2023 AMS Spring Southeastern Sectional Meeting claiming that
it is an impossible proof. They cited a false claim in Loomis’ 1907 book “There are no trigono-
metric proofs. ...... Trigonometry is because the Pythagorean Proposition is.” This note
presents a new method based on similarity and geometric progression with which a pure geo-
metrical proof is given. Additionally, this note also discusses some proofs in Loomis’ book and
provides more new proofs using the concept of the Lemoine Point. Finally, the Appendix has
Zimba’s original proof of the angle difference identities for sin() and cos() without using the
Pythagorean Theorem. It also includes a proof of the angle sum identities for sin() and cos()
without using the Pythagorean Theorem. Both results can be used to prove the Pythagorean
Identity. As a result, Loomis’ claim is false and the proof of Jackson-Johnson can be quickly
replaced by a purely geometrical one.
A proof of the Pythagorean Theorem using trigonometry was presented at the AMS Spring
Southeastern Sectional Meeting on March 18, 2023 by Ne’Kiya D. Jackson and Calcea Rujean
Johnson [4]. This was reported widely by the media such as The Guardian [12], Popular Mechan-
ics [8] and Scientific American [11]. Unfortunately, the authors and some reports kept suggesting
that a trigonometric proof is “impossible.” They all cited a 1907 book The Pythagorean Propo-
sition by Elisha Scott Loomis [6, second edition, pp. 244-245] (Figure 1(a)) in which Loomis
(Figure 1(b)) stated the following:
Facing forward the thoughtful reader may raise the question: Are there any proofs
based upon the science of trigonometry or analytical geometry?
There are no trigonometric proofs, because all the fundamental formulae of trigonom-
etry are themselves based upon the truth of the Pythagorean Theorem; because of this
theorem we say sin
2
A + cos
2
A = 1, etc. Trigonometry is because the Pythagorean
Theorem is [6, p.244].
This is false, because Zimba [13] showed that the validity of cos(α − β) = cos(α) cos(β) +
sin(α)sin(β) and sin(α − β) = sin(α)cos(β) − cos(α) sin(β) is independent of the Pythagorean
Identity
1
which can be proved using cos(α − β) by setting α = β. This author is unaware of other
1
The Pythagorean Identity refers to the identity sin
2
(x) + cos
2
(x) = 1, which implies the Pythagorean Theorem
immediately. Conversely, if the Pythagorean Identity holds, the Pythagorean Theorem can be obtained easily.
1