Full Angle Method

The definition of angles in common geometry textbooks uses the order relation. In unordered geometries, we need to use full-angles that do not involve the order relation. We will presents the definition and the basic properties of full-angles.


R1. A full-angle is defined as an ordered pair of two lines u and v denoted by ∠[u, v].
R2. ∠[u, v] = −∠[v, u].
R3. Constant full-angle ∠[u, u] is denoted by ∠[0]. We have ∠[0] + ∠[u, v] = ∠[u, v] + ∠[0] = ∠[u, v]. Also ∠[u, v] = ∠[0] iff // v.
R4. If u ⊥ v, then we define ∠[u, v] to be a constant full-angle ∠[1]. ∠[u, v] + ∠[u, v] = ∠[0], or ∠[1] + ∠[1] = ∠[0].
R5. For any line l, the operation ”addition” between two full angles is defined as∠[u, v] = ∠[u, l]+∠[l, v]. Thus the addition of two full-angles does not use the order relation and is diagramindependent.


The set of all full-angles with the addition operator “+”, is an additive (Abelian) group.


Remark. The geometries with the definition of the addition of full-angles are Euclidean, as opposed to non-Euclidean geometries. For example, the assertion that the sum of three angles of a triangle ABC equals 180◦, which is equivalent to Euclidean fifth hypothesis, can be expressed as
∠[CA,AB] + ∠[AB,BC] + ∠[BC,CA] = ∠[CA,CA] = ∠[0].


R6. If l ⊥ v, we have ∠[u, v] = ∠[u, l] + ∠[l, v] = ∠[u, l] + ∠[1]. Note that this rule is the combination of Rule4 and Rule5.
R7. If AB = AC, then ∠[AB,BC] = ∠[BC,CA].


The full-angle method works particularly well for theorems involving circles and angles. Let A,B,C, and D be four non-collinear points.5 We use ∠[ABC] to denote the full-angle ∠[AB,BC] and use ∠ABC to denote the traditional angle. In ordered geometry, ∠[ABC] = ∠[DEF] if and only if ∠ABC = ∠DEF and the two angles have the same orientation, or ∠ABC = 180 − ∠DEF and the two angles have the opposite orientations.

The Inscribed Angle Theorem
R8. (The Inscribed Angle Theorem) Points A,B,C, and D are cyclic iff ∠[ABC] = ∠[ADC] (the left of Figure 5.)


Applying this twice we can have a combined rule, say, ∠[BAC] = ∠[BDC] iff ∠[ABC] = ∠[ADC] (the right of Figure 5), etc. Here the four points can be in any order. These diagram-independent properties of full-angles make the proofs not only much simpler and
diagram-independent, but also rigorous.


Example 6. The same as Example 1. Let points C and D be the intersections of two circles (A) and (B), line HF be passing through C and meeting (A) and (B) at H and F, line GE be passing through D and meeting (A) and (B) at G and E. Show that GH is parallel to EF (Figure 6).

A Two Circle Theorem


Proof
∠[GH,EF] = ∠[GHC] + ∠[CFE] (1)
(∵ ∠[GHC] = ∠[GDC] and ∠[CFE] = ∠[CDE]) (2)
= ∠[GDC] + ∠[CDE] (3)
= ∠[GDE] = ∠[0] (4)


In this way we prove that ∠[GH,EF] = ∠[0], thus we have GH // EF from R3. In Step (2), the two angle equality assertions use R8 with the hypotheses cyclic(HDGC) and cyclic(CFED). This gives us a way to organize the proof in a hierarchical way: by clicking ‘∵ ∠[GHC] = ∠[GDC]’, it expands to include a substep ‘∵ cyclic(HDGC)’ on which the rule is applied; by clicking cyclic(HDGC) it might expand a substep for the proof of the assertion cyclic(HDGC)
and in this example it is not needed to prove because it is a hypothesis.

Here we deliberately use the order HDGC instead of the clockwise order HGDC to emphasize the diagram-independent nature of the full-angle proof. This proof is a diagram-independent proof.


Also here we implicitly use hypotheses on collinear facts, e.g., in Step (1), we use the fact that ∠[GHC] is ∠[GHF] and this fact is true because H, C, and F are collinear, etc. The proof adds an auxiliary segment CD. This addition is natural for our full-angle method because the proof requires ∠[GDC], hence requires segment CD. We will discuss the topic on adding auxiliary geometric elements for the full-angle method.

For detail please read our paper >>

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