Proof: If Three Sides Ratios Equal Then Similar Triangles

Let's prove the following theorem:

if (distance AB) / (distance XY) = (distance BC) / (distance YZ) and (distance BC) / (distance YZ) = (distance CA) / (distance ZX) and distance AC = distance SZ and ST || XY and m∠XSZ = 180 and m∠YTZ = 180, then △ABC ∼ △XYZ

Y Z X S T A C B

Proof:

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Given
1 (distance AB) / (distance XY) = (distance BC) / (distance YZ)
2 (distance BC) / (distance YZ) = (distance CA) / (distance ZX)
3 distance AC = distance SZ
4 ST || XY
5 m∠XSZ = 180
6 m∠YTZ = 180
Proof Table
# Claim Reason
1 XY || ST if ST || XY, then XY || ST
2 m∠TSZ = m∠YXS if XY || ST and m∠XSZ = 180, then m∠TSZ = m∠YXS
3 m∠YXS = m∠YXZ if m∠XSZ = 180, then m∠YXS = m∠YXZ
4 m∠TSZ = m∠YXZ if m∠TSZ = m∠YXS and m∠YXS = m∠YXZ, then m∠TSZ = m∠YXZ
5 m∠YXZ = m∠TSZ if m∠TSZ = m∠YXZ, then m∠YXZ = m∠TSZ
6 m∠XZY = m∠SZT if m∠XSZ = 180 and m∠YTZ = 180, then m∠XZY = m∠SZT
7 △XZY ∼ △SZT if m∠YXZ = m∠TSZ and m∠XZY = m∠SZT, then △XZY ∼ △SZT
8 △XYZ ∼ △STZ if △XZY ∼ △SZT, then △XYZ ∼ △STZ
9 (distance ZS) / (distance ZX) = (distance ST) / (distance XY) if △XZY ∼ △SZT, then (distance ZS) / (distance ZX) = (distance ST) / (distance XY)
10 distance ZS = distance CA if distance AC = distance SZ, then distance ZS = distance CA
11 distance CA = distance ZS if distance ZS = distance CA, then distance CA = distance ZS
12 (distance CA) / (distance ZX) = (distance ST) / (distance XY) if (distance ZS) / (distance ZX) = (distance ST) / (distance XY) and distance CA = distance ZS, then (distance CA) / (distance ZX) = (distance ST) / (distance XY)
13 (distance AB) / (distance XY) = (distance CA) / (distance ZX) if (distance AB) / (distance XY) = (distance BC) / (distance YZ) and (distance BC) / (distance YZ) = (distance CA) / (distance ZX), then (distance AB) / (distance XY) = (distance CA) / (distance ZX)
14 (distance ST) / (distance XY) = (distance AB) / (distance XY) if (distance AB) / (distance XY) = (distance CA) / (distance ZX) and (distance CA) / (distance ZX) = (distance ST) / (distance XY), then (distance ST) / (distance XY) = (distance AB) / (distance XY)
15 distance ST = distance AB if (distance ST) / (distance XY) = (distance AB) / (distance XY), then distance ST = distance AB
16 distance AB = distance ST if distance ST = distance AB, then distance AB = distance ST
17 (distance ST) / (distance XY) = (distance TZ) / (distance YZ) if △XZY ∼ △SZT, then (distance ST) / (distance XY) = (distance TZ) / (distance YZ)
18 (distance ST) / (distance XY) = (distance BC) / (distance YZ) if (distance ST) / (distance XY) = (distance AB) / (distance XY) and (distance AB) / (distance XY) = (distance BC) / (distance YZ), then (distance ST) / (distance XY) = (distance BC) / (distance YZ)
19 (distance TZ) / (distance YZ) = (distance BC) / (distance YZ) if (distance ST) / (distance XY) = (distance TZ) / (distance YZ) and (distance ST) / (distance XY) = (distance BC) / (distance YZ), then (distance TZ) / (distance YZ) = (distance BC) / (distance YZ)
20 distance BC = distance TZ if (distance TZ) / (distance YZ) = (distance BC) / (distance YZ), then distance BC = distance TZ
21 △ABC ≅ △STZ if distance AB = distance ST and distance BC = distance TZ and distance CA = distance ZS, then △ABC ≅ △STZ
22 △ABC ∼ △STZ if △ABC ≅ △STZ, then △ABC ∼ △STZ
23 △ABC ∼ △XYZ if △ABC ∼ △STZ and △XYZ ∼ △STZ, then △ABC ∼ △XYZ

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