A New Proof of the Pythagorean Theorem and Its Application to Element Decompositions in Topological Algebras

Figure 1

Diagram relating to proof that the function giving the length of a line segment with one endpoint at 𝑜 has continuous directional derivatives at points other than 𝑜, Lemma 2.3. Three facts used in the proof are as follows: (1) Δ{𝑝,𝑜,𝑠⊕𝜖𝑠𝑤} is similar to Δ{𝑝,𝑠⊕𝜖𝑠𝑤,𝑟} because ∠{𝑜,𝑠⊕𝜖𝑠𝑤,𝑟} is a right angle, (2) for all 𝜖≠0, the triangles Δ{𝑝,𝑠,𝑠⊕𝜖𝑠𝑤} are similar, (3) 𝑞 is between 𝑝 and 𝑟.