Enter the length of lines A to C, C to E, and A to B from the diagram into the calculator to determine the length of B to D using the side-splitter theorem (note: A to B is the full side, so A to D = A to B − B to D).

Side Splitter Theorem Calculator
Leave one field blank to solve. Use either the split-segments form or the whole-side form.
Split Segments
Whole Side
Uses the standard side-splitter proportion: AD / DB = AE / EC
Uses the whole-side proportion: AD / AB = AE / AC

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.

Side Splitter Theorem Formula

\frac{AC}{CE} = \frac{AB}{BD}

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
BD 
BD = \frac{AB \cdot CE}{AC}
CE 
CE = \frac{AC \cdot BD}{AB}
AB 
AB = \frac{AC \cdot BD}{CE}
AC 
AC = \frac{AB \cdot CE}{BD}

What the Variables Mean

  • AB and BD are the two split segments on one side of the triangle.
  • AC and CE are the two split segments on the other side of the triangle.
  • AB corresponds to AC, and BD corresponds to CE.

How to Use the Calculator

  1. Identify the three known segment lengths from the triangle.
  2. Make sure all measurements use the same unit before entering them.
  3. Leave the unknown value blank.
  4. Calculate the missing segment using the proportional relationship.

Example

If AB = 5, BD = 4, and AC = 3, the missing value CE is:

CE = \frac{AC \cdot BD}{AB}
CE = \frac{3 \cdot 4}{5} = 2.4

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

  • The interior segment must be parallel to the third side of the triangle.
  • 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.