How this calculator works
A parallel segment creates similar triangles ADE and ABC. Their corresponding whole-side ratios are equal, and subtracting each inner part gives the equivalent split-segment proportion AD/DB = AE/EC. Choose any missing segment to rearrange this proportion. Check mode applies the converse: equal split ratios place D and E at the same fraction along their respective sides, so DE is parallel to BC. The tool assumes straight sides, distinct vertices and interior division points. Numerical equality uses a relative tolerance of 1e-10.
Formula
AD/DB = AE/EC; AD × EC = AE × DB; k = AD/(AD+DB) = AE/(AE+EC) = DE/BC.
Worked example
If AD = 2 m, DB = 3 m and AE = 4 m, then EC = 4 × 3 / 2 = 6 m. The whole sides are AB = 5 m and AC = 10 m. The inner triangle has length scale 2/5 = 0.4.
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Frequently asked questions
Do I compare parts or whole sides?
Both work when used consistently: AD/DB equals AE/EC, while AD/AB equals AE/AC. Mixing a split denominator with a whole-side denominator is incorrect.
Can a point lie beyond the triangle?
This tool uses positive interior segments only. External division requires directed lengths and a separate sign convention.
Why is scale omitted when the converse fails?
Unequal ratios mean the smaller triangle does not share a common length scale with the whole triangle under the stated correspondence.