Use the side splitter theorem to find a missing triangle segment (AD, DB, AE or EC) from the other three, or check whether DE is parallel to BC.
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Understanding the Side Splitter Theorem Calculator
The side splitter theorem describes a proportional relationship inside a triangle. When a segment is drawn parallel to one side of a triangle, it divides the other two sides into matching parts that share the same ratio. This calculator uses that relationship to solve for a missing segment length when any three of the four values are known. For the stated triangle arrangement, the converse check compares all four split lengths. Unequal ratios within its 0.1% relative tolerance are labelled approximate only; rounded measurements and tolerance do not prove exact parallelism.
Side Splitter Theorem Formula
This means the ratio of the two segments on one side of the triangle equals the ratio of the corresponding two segments on the other side.
Rearranged Forms
You can solve for any missing variable by rearranging the same proportion.
| Unknown Length | Formula |
|---|---|
| DB | DB = (AD × EC) / (AE) |
| EC | EC = (DB × AE) / (AD) |
| AD | AD = (DB × AE) / (EC) |
| AE | AE = (AD × EC) / (DB) |
What the Variables Mean
- AD and DB split side AB, with D between A and B.
- AE and EC split side AC, with E between A and C.
- AD corresponds to AE, and DB corresponds to EC.
How to Use the Calculator
- Select AD, DB, AE or EC as the missing segment, then identify the other three split lengths.
- Make sure all measurements use the same unit before entering them.
- Enter the three displayed lengths; the selected unknown is hidden. The converse check instead requires all four split lengths.
- Calculate the missing segment using the proportional relationship.
Example
If AD = 5, DB = 4, and AE = 3, with DE parallel to BC, the missing value EC is:
So the missing segment length is 2.4 in the same unit used for the other measurements.
Why the Theorem Works
The theorem comes from triangle similarity. A line drawn parallel to one side of a triangle creates a smaller triangle inside the original one. Because the two triangles are similar, their corresponding sides are proportional. The side splitter theorem is simply that similarity rule written in terms of the split side lengths.
Conditions Required
- Finding a missing length assumes DE is parallel to BC. For the converse check, D must be inside AB and E inside AC.
- All four lengths must come from the same triangle configuration.
- Measurements must be positive values.
- Units must be consistent throughout the calculation.
Common Mistakes
- Using mixed units such as inches for one segment and centimeters for another.
- Pairing the wrong corresponding segments in the proportion.
- Using a diagram where the interior segment is not actually parallel to the opposite side.
- Entering a full side length when the formula requires only one split portion of that side.
Helpful Tips
- If you know a whole side length instead of one split segment, subtract the other known part first.
- Check that the larger segment pair and smaller segment pair appear in the same positions in your ratio.
- If the result seems unreasonable, verify both the diagram labels and the unit conversions.
Where This Theorem Is Used
- Geometry classes and similar triangle proofs
- Scale drawings and technical sketches
- Construction layout and measurement checks
- Surveying and indirect distance estimation
Quick Answers
Can this calculator solve for any one of the four lengths?
Yes. As long as three values are known, the fourth can be found from the same proportional equation.
Do the units matter?
Any length unit can be used, but every entered value must be in the same unit.
What if the line is not parallel to the triangle's base?
Then the side splitter theorem does not apply, and the proportion will not generally be valid.
