## Jan Stevens' proof of E. W. Dijkstra's Generalization

If, in a ΔABC, angles α, β, γ lie opposite the sides of length a, b, c, then

(*) | sign(α + β - γ) = sign(a² + b² - c²), |

In the proof of his generalization, Dijkstra ran into difficulties with geometric illustrations. As he himself wrote: Note that I carefully avoided the pictures for

α + β > γ | a² + b² > c². |

The diagrams make the construction obvious. ABKH is an isosceles trapezoid that degenerates in rectangle for the right angled triangle (see proof #41.) Dijkstra's formula follows from the following

### Lemma

In an isosceles trapezoid, of the two bases, the base incident to smaller (base) angles is longer, and vice versa. |

### Proof

According to the Fifth Postulate, two lines traversed by a third meet on that side of the transversal for which the sum of the internal angles is less than two right angles (180°). This means that the point of intersection of the side lines AH and BK (say P) is closer to AB if

a² + b² > c² iff α + β > γ, |

or, which is the same if we exclude the equality case as in proof #41,

a² + b² < c² iff α + β < γ, |

Dijkstra's identity is the combination of these three cases. In fact, there is no need to invoke proof #41. Indeed, since

There is more than one way of handling the isosceles trapezoid ABKH. For example, draw the diagonal BH and consider two triangles ABH and KHB. The triangles have a pair of equal sides: AH = BK, with BH being common to both triangles. According to Euclid I.24, of the third sides that is bigger which subtends a bigger angle. (Euclid I.25 proves the converse.) However, in the two triangles,

∠ABH = ∠BHK, |

so that

Jan Stevens actually proves more, viz.,

a² + b² - c² = 2ab sin( (α + β - γ)/2 ). |

In case where α + β > γ the proof is based on the following diagram

where AE and BF are perpendicular to HK. Since

In case α + β < γ, the derivation is similar. The case of the right-angled triangle fits into the formula as well and can be referred to Proof 41.

The formula can be rewritten as

a² + b² - c² = 2ab cos γ, |

giving an additional proof of the Cosine Law.

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