Proof of Theorem 1.3
The midpoint of a line segment is unique. If P = (a, b) and Q = (c, d) are endpoints of a line segment,
,
then the midpoint of PQ is
.
If P = (a, b) and Q = (c, d) are endpoints of a line segment, then the midpoint of PQ is
,
.
Postulate 1.10: Given collinear points P, Q, and R; for PQ with length m and QR with length n, the length
of PQR is m+n.
Definition: For collinear points P, Q, R; Q is a midpoint of P and R if it is equal distance from P and Q (m is
congruent to n).
The length of a segment is given by postulate 1.11 For P = (a,b) and R = (c,d),
𝑃𝑅
a-c
a-c
b-d
b-d
The length m of PQ is
1
2
1
𝑃𝑅
2
m
a-c
1
4
a-c
4
b-d
4
b-d
a-c
b-d
4
a-c
2
b-d
2
By postulate 1.3, we can assign P = (0,0).
0-c
2
0-d
2
c
2
d
2
c = c + 0, and d = d + 0, therefore
c
0
d
2
0
a
2
c
b
2
d
2
Similarly,
n
1
RP
2
1
2
c-a
c-a
2
d-b
d-b
2
By postulate 1.3,
0-c
2
0-d
2
Therefore, the midpoint of PQ is
c
2
d
2
,
c
0
2
.
d
0
2
a
c
2
b
d
2
The midpoint of a line segment is unique.
Suppose not.
Let m1 and m2 be two distinct midpoints for the line segment PQ, with m1 ≠ m2. By the definition of the
,
midpoint of a segment PQ proved above, m1 =
and m2 =
Since m1 and m2 are distinct, x1 ≠ x2 and y1 ≠ y2.
x1 =
𝑎
, x2 =
𝑐
𝑎
𝑐
2
2
Contradiction
y1 =
b
d
2
Contradiction
Therefore, the midpoint is unique.
, x2 =
b
d
2
,
.