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PROOF 108

Prove.

The midpoint formula

On the coordinate plane.

What you may use

A1d(P,Q) = √(Δx² + Δy²)
distance between two points (Proof 21, cited)
A2P = A + t(B−A), 0 ≤ t ≤ 1
point on segment AB
A3t ≥ 0 ⟹ √(t²k) = t√k
square roots and positive multiples
A4a(b+c) = ab + ac
distributive law
A5a + b = b + a
commutativity of addition
A6(a+b)+c = a+(b+c)
associativity of addition
A7a + (−a) = 0, a + 0 = a
additive inverse and identity
A8ab = ba, (ab)c = a(bc)
commutativity and associativity of multiplication
A9a × 1 = a
multiplicative identity
A10a ≠ 0 ⟹ a(1/a) = 1
multiplicative inverse
A11ca = cb, c ≠ 0 ⟹ a = b
cancellation law of multiplication
A12A ≠ B ⟹ AB ≠ 0
distance between two distinct points
A130 ≤ t ≤ 1 ⟹ 0 ≤ 1 − t
order
A140 < ½ < 1, 2 ≠ 0
position of ½ and 2
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

For two distinct points A, B, the midpoint M of segment AB is the point that lies on segment AB and satisfies AM = MB.

Proof.

Related labs

The Coordinate Plane lab →