Definition D1
The line ℓ(m, b) is the set of all points (x, y) satisfying y = mx + b. That is, (x, y) ∈ ℓ(m, b) ⟺ y = mx + b
Proof.
Vertical lines are not treated here. Both lines are assumed to have a slope.convention
∎
Definition D2
Two distinct lines are said to be parallel when they have no point in common.
Proof.
(1)Parallelism is defined not by a picture but by “they do not meet.”convention
(2)Two coincident lines are not distinct, so they are excluded from this definition.convention
∎
Lemma L1
If m₁ ≠ m₂, then ℓ(m₁, b₁) and ℓ(m₂, b₂) have a common point.
Proof.
(1)m₁ ≠ m₂, so m₁ − m₂ ≠ 0assumption, A8
(2)Let s = 1/(m₁ − m₂). Then (m₁ − m₂)s = 1(1), A6
(3)Let x₀ = (b₂ − b₁)s, y₀ = m₁x₀ + b₁.(2)
(4)(x₀, y₀) ∈ ℓ(m₁, b₁)(3), D1
(5)(m₁ − m₂)x₀ = (b₂ − b₁)((m₁ − m₂)s) = (b₂ − b₁) × 1 = b₂ − b₁(2), (3), A4, A5
(6)(m₁ − m₂)x₀ = m₁x₀ − m₂x₀, so m₁x₀ − m₂x₀ = b₂ − b₁(5), A1, A7
(7)Adding m₂x₀ + b₁ to both sides gives m₁x₀ + b₁ = m₂x₀ + b₂(6), E, A2, A3
(8)y₀ = m₂x₀ + b₂, so (x₀, y₀) ∈ ℓ(m₂, b₂)(3), (7), D1
(9)Therefore the two lines meet at (x₀, y₀).(4), (8)
∎
Theorem T1
If two lines are parallel, their slopes are equal.
Proof.
(1)The contrapositive of L1 is: if the two lines do not meet, then m₁ = m₂.L1, A11
(2)Parallel lines have no common point.D2
(3)Therefore, if the lines are parallel, m₁ = m₂(1), (2)
∎
Theorem T2
If m₁ = m₂ and b₁ ≠ b₂, the two lines are parallel.
Proof.
(1)Let m = m₁ = m₂.assumption
(2)Suppose there is a common point (x, y).assumption
(3)y = mx + b₁ and y = mx + b₂(1), (2), D1
(4)mx + b₁ = mx + b₂, so b₁ = b₂(3), A9
(5)This contradicts b₁ ≠ b₂.(4), assumption
(6)Therefore there is no common point.(5)
(7)m × 0 + b₁ = b₁, so (0, b₁) lies only on ℓ(m, b₁), and the two lines are distinct.(1), assumption, A10, A3, D1
(8)Therefore the two lines are parallel.(6), (7), D2
∎
Corollary C
There are only three possible relations between two lines. They meet at one point, they are parallel, or they are the same line.
Proof.
(1)If m₁ ≠ m₂, they meet, and at exactly one point.L1, A12
(2)If m₁ = m₂ and b₁ ≠ b₂, they are parallel.T2
(3)If m₁ = m₂ and b₁ = b₂, the two sets are equal, so they are the same line.D1
(4)These three cases cover every case.(1), (2), (3)
∎