Reading the split backwards
Two equal split ratios can show that a line is parallel. No angle is needed.
- D on AB: AD is 4, DB is 6.
- E on AC: AE is 6, EC is 9. DE runs parallel to BC.
- F on AC instead: AF is 3, FC is 12. DF tilts.
- Equal splits give parallel. Unequal splits give a tilt.
The converse
The theorem goes from "parallel" to "equal ratios". The converse goes the other way.
A line meets two sides of a triangle. It cuts both in the same ratio, counted from the same corner. Then the line is parallel to the third side. If the ratios differ, it is not parallel.
Why? Draw a line through D parallel to BC. By the theorem it cuts AC in the same ratio. Only one point of AC gives that ratio. So that point is E, and DE is the parallel line.
Check two lines
Take D on AB with AD = 4 and DB = 6. Take E and F on AC. Then AE = 6, EC = 9, AF = 3 and FC = 12.
Line DE gives AD ÷ DB = 4/6 and AE ÷ EC = 6/9. Both are 2/3. So DE is parallel to BC.
Line DF gives AD ÷ DB = 4/6 and AF ÷ FC = 3/12. That is 2/3 against 1/4. They differ. So DF is not parallel to BC.
Watch out!
Do not compare AD ÷ AB with AE ÷ EC. AB is a whole side and EC is a piece. Compare piece with piece, or whole with whole.
Check six lines.
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