Equality Example
Transitive Property of Equality Variation 1
Divide Both Sides
Multiply Substitute Two Terms
Multiply Substitute Two Terms 2
Zero Plus a
Transitive Property of Equality Variation 2
Multiplicative Property of Equality Variation 2
Transitive Property of Equality Variation 3
Division is Commutative
Associative Property
Multiplicative Property of Equality Variation 1
Simplify
Transitive
Distributive Property 2
Commutative Property Variation 1
Substitution 2
Substitution 6
Distributive Property 3
Associative Property of Multiplication 2
Transitive Property Application 2
Substitution 10
Substitution 12
Divide Simplify 2
Simplify 2
Simplify 3
Transitive 2
Subtract Same Sides
Divide Both
Divide Substitute 2
Divide Substitute 4
Slope 1
Subtract Zero Example
Subtraction Example
Subtract to Zero
Example 2
Subtract Associative
Subtract 1
Divide Zero 2
Substitute Two Terms
Subtract Zero Example 2
Transitive Subtract 3
Substitution 13
Midsegment Triangle 2

Proof: Midsegment Triangle 2

Let's prove the following theorem:

if X is the midpoint of line RT and Y is the midpoint of line ST, then (distance XY) ⋅ 2 = distance RS

S T R X Y

Proof:

View as a tree | View dependent proofs | Try proving it

Given
1 X is the midpoint of line RT
2 Y is the midpoint of line ST
Assumptions
3 the x coordinate of point R = 0
4 the y coordinate of point R = 0
5 the x coordinate of point S = b ⋅ 2
6 the y coordinate of point S = 0
7 the x coordinate of point T = a ⋅ 2
8 the y coordinate of point T = c ⋅ 2
9 b > 0
Proof Table
# Claim Reason
1 the x coordinate of point X = ((the x coordinate of point R) + (the x coordinate of point T)) / 2 if X is the midpoint of line RT, then the x coordinate of point X = ((the x coordinate of point R) + (the x coordinate of point T)) / 2
2 the x coordinate of point X = (0 + (a ⋅ 2)) / 2 if the x coordinate of point R = 0 and the x coordinate of point T = a ⋅ 2 and the x coordinate of point X = ((the x coordinate of point R) + (the x coordinate of point T)) / 2, then the x coordinate of point X = (0 + (a ⋅ 2)) / 2
3 the x coordinate of point X = a if the x coordinate of point X = (0 + (a ⋅ 2)) / 2, then the x coordinate of point X = a
4 the y coordinate of point X = ((the y coordinate of point R) + (the y coordinate of point T)) / 2 if X is the midpoint of line RT, then the y coordinate of point X = ((the y coordinate of point R) + (the y coordinate of point T)) / 2
5 the y coordinate of point X = (0 + (c ⋅ 2)) / 2 if the y coordinate of point R = 0 and the y coordinate of point T = c ⋅ 2 and the y coordinate of point X = ((the y coordinate of point R) + (the y coordinate of point T)) / 2, then the y coordinate of point X = (0 + (c ⋅ 2)) / 2
6 the y coordinate of point X = c if the y coordinate of point X = (0 + (c ⋅ 2)) / 2, then the y coordinate of point X = c
7 the x coordinate of point Y = ((the x coordinate of point S) + (the x coordinate of point T)) / 2 if Y is the midpoint of line ST, then the x coordinate of point Y = ((the x coordinate of point S) + (the x coordinate of point T)) / 2
8 the x coordinate of point Y = ((b ⋅ 2) + (a ⋅ 2)) / 2 if the x coordinate of point S = b ⋅ 2 and the x coordinate of point T = a ⋅ 2 and the x coordinate of point Y = ((the x coordinate of point S) + (the x coordinate of point T)) / 2, then the x coordinate of point Y = ((b ⋅ 2) + (a ⋅ 2)) / 2
9 the x coordinate of point Y = b + a if the x coordinate of point Y = ((b ⋅ 2) + (a ⋅ 2)) / 2, then the x coordinate of point Y = b + a
10 the y coordinate of point Y = ((the y coordinate of point S) + (the y coordinate of point T)) / 2 if Y is the midpoint of line ST, then the y coordinate of point Y = ((the y coordinate of point S) + (the y coordinate of point T)) / 2
11 the y coordinate of point Y = (0 + (c ⋅ 2)) / 2 if the y coordinate of point S = 0 and the y coordinate of point T = c ⋅ 2 and the y coordinate of point Y = ((the y coordinate of point S) + (the y coordinate of point T)) / 2, then the y coordinate of point Y = (0 + (c ⋅ 2)) / 2
12 the y coordinate of point Y = c if the y coordinate of point Y = (0 + (c ⋅ 2)) / 2, then the y coordinate of point Y = c
13 slope of line RS = ((the y coordinate of point S) - (the y coordinate of point R)) / ((the x coordinate of point S) - (the x coordinate of point R)) slope of line RS = ((the y coordinate of point S) - (the y coordinate of point R)) / ((the x coordinate of point S) - (the x coordinate of point R))
14 slope of line RS = (0 - 0) / ((b ⋅ 2) - 0) if slope of line RS = ((the y coordinate of point S) - (the y coordinate of point R)) / ((the x coordinate of point S) - (the x coordinate of point R)) and the y coordinate of point S = 0 and the y coordinate of point R = 0 and the x coordinate of point S = b ⋅ 2 and the x coordinate of point R = 0, then slope of line RS = (0 - 0) / ((b ⋅ 2) - 0)
15 (0 - 0) / ((b ⋅ 2) - 0) = 0 if b > 0, then (0 - 0) / ((b ⋅ 2) - 0) = 0
16 slope of line RS = 0 if slope of line RS = (0 - 0) / ((b ⋅ 2) - 0) and (0 - 0) / ((b ⋅ 2) - 0) = 0, then slope of line RS = 0
17 slope of line XY = ((the y coordinate of point Y) - (the y coordinate of point X)) / ((the x coordinate of point Y) - (the x coordinate of point X)) slope of line XY = ((the y coordinate of point Y) - (the y coordinate of point X)) / ((the x coordinate of point Y) - (the x coordinate of point X))
18 slope of line XY = (c - c) / ((b + a) - a) if slope of line XY = ((the y coordinate of point Y) - (the y coordinate of point X)) / ((the x coordinate of point Y) - (the x coordinate of point X)) and the y coordinate of point Y = c and the y coordinate of point X = c and the x coordinate of point Y = b + a and the x coordinate of point X = a, then slope of line XY = (c - c) / ((b + a) - a)
19 (c - c) / ((b + a) - a) = 0 if b > 0, then (c - c) / ((b + a) - a) = 0
20 slope of line XY = 0 if slope of line XY = (c - c) / ((b + a) - a) and (c - c) / ((b + a) - a) = 0, then slope of line XY = 0
21 distance RS = (the x coordinate of point S) - (the x coordinate of point R) if slope of line RS = 0, then distance RS = (the x coordinate of point S) - (the x coordinate of point R)
22 distance RS = (b ⋅ 2) - 0 if the x coordinate of point S = b ⋅ 2 and the x coordinate of point R = 0 and distance RS = (the x coordinate of point S) - (the x coordinate of point R), then distance RS = (b ⋅ 2) - 0
23 distance RS = b ⋅ 2 if distance RS = (b ⋅ 2) - 0, then distance RS = b ⋅ 2
24 distance XY = (the x coordinate of point Y) - (the x coordinate of point X) if slope of line XY = 0, then distance XY = (the x coordinate of point Y) - (the x coordinate of point X)
25 distance XY = (b + a) - a if the x coordinate of point Y = b + a and the x coordinate of point X = a and distance XY = (the x coordinate of point Y) - (the x coordinate of point X), then distance XY = (b + a) - a
26 distance XY = b if distance XY = (b + a) - a, then distance XY = b
27 distance RS = (distance XY) ⋅ 2 if distance XY = b and distance RS = b ⋅ 2, then distance RS = (distance XY) ⋅ 2
28 (distance XY) ⋅ 2 = distance RS if distance RS = (distance XY) ⋅ 2, then (distance XY) ⋅ 2 = distance RS
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