Commutative Property Variation 1

Multiplicative Identity 2

Distributive Property 4

Multiplicative Property of Equality Variation 1

Addition Theorem

Midpoint Distance

Proof: Midpoint Distance

Let's prove the following theorem:

if M is the midpoint of line AB, then distance AB = (distance AM) ⋅ 2

Proof:

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Given

1	M is the midpoint of line AB

Proof Table

#	Claim	Reason
1	(distance AM) + (distance MB) = distance AB	if M is the midpoint of line AB, then (distance AM) + (distance MB) = distance AB (Midpoint Sum)
2	distance AB = (distance AM) + (distance MB)	if (distance AM) + (distance MB) = distance AB, then distance AB = (distance AM) + (distance MB) (Symmetric Property of Equality)
3	distance AM = distance MB	if M is the midpoint of line AB, then distance AM = distance MB (Midpoint Property)
4	distance AB = (distance AM) + (distance AM)	if distance AB = (distance AM) + (distance MB) and distance AM = distance MB, then distance AB = (distance AM) + (distance AM) (Substitution 3)
5	distance AB = (distance AM) ⋅ 2	if distance AB = (distance AM) + (distance AM), then distance AB = (distance AM) ⋅ 2 (Sum to Double)

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