Geometry (Beta) / Chapter 6: Similar Triangles / Similar Triangles

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

Proof:

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Given

1	X is the midpoint of line RT
2	Y is the midpoint of line ST

Additional 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

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#	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 (Midpoint is Halfway)
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 (Multiply Substitute Two Terms 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 (transitive)
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 (Midpoint is Halfway 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 (Multiply Substitute Two Terms 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 (transitive)
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 (Midpoint is Halfway)
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 (Multiply Substitute Two Terms 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 (Transitive 2)
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 (Midpoint is Halfway 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 (Multiply Substitute Two Terms 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 (transitive)
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)) (Slope of a Line)
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) (Slope 1)
15	(0 - 0) / ((b ⋅ 2) - 0) = 0	if b > 0, then (0 - 0) / ((b ⋅ 2) - 0) = 0 (Example 2)
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 (Transitive Property of Equality)
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)) (Slope of a Line)
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) (Slope 1)
19	(c - c) / ((b + a) - a) = 0	if b > 0, then (c - c) / ((b + a) - a) = 0 (Divide Zero 2)
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 (Transitive Property of Equality)
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) (Slope Property)
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 (Substitute Two Terms)
23	distance RS = b ⋅ 2	if distance RS = (b ⋅ 2) - 0, then distance RS = b ⋅ 2 (Subtract Zero Example 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) (Slope Property)
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 (Substitute Two Terms)
26	distance XY = b	if distance XY = (b + a) - a, then distance XY = b (Transitive Subtract 3)
27	distance RS = (distance XY) ⋅ 2	if distance XY = b and distance RS = b ⋅ 2, then distance RS = (distance XY) ⋅ 2 (Substitution 13)
28	(distance XY) ⋅ 2 = distance RS	if distance RS = (distance XY) ⋅ 2, then (distance XY) ⋅ 2 = distance RS (Symmetric Property of Equality)