Triangle Proportionality Theorem Calculator - AD/DB = AE/EC Solver

The triangle proportionality theorem calculator solves AD / DB = AE / EC for any one of the four segments and reports the constant of proportionality k, the full side AB = AD + DB, and the full side AC = AE + EC. Pick the unknown with the Find toggle, enter the other three segments in the same unit, and the result updates as you type.

• • Geometry homework and proofs: Use AD / DB = AE / EC to find the missing segment from three given pieces.
• • Finding the parallel segment DE: Combine the theorem with AA similarity to get DE from AD, DB, and BC via DE / BC = AD / AB.
• • Midsegment and median constructions: When AD = DB and AE = EC the line DE is the midsegment, so k = 1 and DE = BC / 2.
• • Cross-section and slope problems: Convert a measured cut on one side of a triangular profile into the matching cut on the other.

The triangle proportionality theorem (also called the basic proportionality theorem, the side-splitter theorem, and Thales' theorem) states that any line DE parallel to side BC of triangle ABC, with D on AB and E on AC, divides the two sides in the same ratio: AD / DB = AE / EC, which also rewrites as AB / AD = AC / AE since AB = AD + DB and AC = AE + EC.

Pick a unit once and keep it. The proportion works in inches, centimeters, meters, or any single length unit; mixing units gives a result that satisfies the algebra yet is useless for any real measurement.

The proportion AD / DB = AE / EC is the segment form of the same similarity result, so the Similar Triangles Calculator is the natural next tool when you also need the scale factor and the area ratio.

The triangle proportionality theorem calculator applies AD / DB = AE / EC to the segment the Find toggle marks as unknown. Once that segment is known, it returns the constant of proportionality k = AD / DB, the full side AB = AD + DB, and the full side AC = AE + EC. The default mode solves Segment DB from AD, AE, and EC.

• AD, DB: Two pieces of side AB. AD runs from A to D; DB runs from D to B. Together they sum to AB.
• AE, EC: Two pieces of side AC. AE runs from A to E; EC runs from E to C. Together they sum to AC.
• k (constant of proportionality): Shared ratio AD / DB = AE / EC, unitless because both numerator and denominator share the same length unit.

The Find toggle controls which segment is solved. Setting Find to Segment AD gives AD = (DB * AE) / EC; Segment DB gives DB = (AD * EC) / AE; Segment AE gives AE = (AD * EC) / DB; Segment EC gives EC = (DB * AE) / AD. After the missing segment is known, k = AD / DB is computed for every mode because both AD and DB are now numbers.

According to Wikipedia's Intercept theorem article, the result is also known as Thales's theorem, the basic proportionality theorem, or the side-splitter theorem, and any line drawn parallel to one side of a triangle divides the other two sides proportionally.

When k = 1 the line DE is the midsegment of triangle ABC and DE = BC / 2, the case the Midsegment of a Triangle Calculator solves directly.

These four ideas decide how the proportion behaves and how to read the ratio k, AB, and AC.

AA similarity is the standard proof. DE parallel to BC makes angle ADE congruent to angle ABC and angle AED congruent to angle ACB, so triangle ABC ~ triangle ADE. The side ratios AD / AB = AE / AC and DE / BC carry through, and the algebra collapses to AD / DB = AE / EC.

Once the proportion has given you the missing segment, the Triangle Calculator extends the same triangle to a full solution with angles, perimeter, and inradius from the three side lengths.

Pick the segment to solve for, enter the three known segments, and read the four derived values.

1. 1 Choose which segment is the unknown: Set the Find dropdown to Segment AD, DB, AE, or EC; the default is Segment DB.
2. 2 Enter the three known segments: Type the other three segments in the same length unit and leave the unknown at zero.
3. 3 Read the missing segment and k: The missing segment is the primary result. k = AD / DB = AE / EC; k = 1 means midpoints, k > 1 means closer to B and C.
4. 4 Read AB and AC: AB = AD + DB and AC = AE + EC are reported so you can verify AB / AD = AC / AE.

A surveyor mapping a slope cross-section as triangle ABC records a reference stake 8 m up AB from A and the parallel reference line meeting AC 12 m up from A. With AD = 8, AE = 12, and EC = 16 from those offsets, Find set to Segment DB returns DB = (8 * 16) / 12 = 10.67 m, AB = 18.67 m, and k = 0.75, so the surveyed side AB is 18.67 m long and the same k carries into the side-splitter ratio DE / BC = AD / AB once BC is measured separately.

If the problem hands you a ratio instead of two segments, the Ratio Calculator turns the ratio into the actual AD and DB values the proportion expects.

Putting the proportion, the ratio k, and both full sides in one calculator keeps the algebra straight for students and surveyors who would otherwise juggle three equations.

• • Solve for any of the four segments: The Find toggle turns AD / DB = AE / EC into four problems.
• • Three derived values in one pass: The ratio k, AB, and AC update with the missing segment.
• • Honest input checks: Zero, negative, and non-numeric inputs are caught with a clear message.
• • Works in any length unit: Inches, feet, centimeters, and meters all work because the proportion only requires shared units.

The constant of proportionality k is the single number a geometry teacher is most likely to ask for after a missing-segment problem. Reporting k together with AB and AC lets a student switch between AD / DB = AE / EC and AB / AD = AC / AE without redoing the arithmetic.

When the four segments are confirmed, the Triangle Side Calculator uses the three resulting sides to compute angles, area, and perimeter in one extra pass.

The proportion is exact, but the inputs and the geometric setup decide how trustworthy the result is.

• • The calculator does not check that the four segments form a valid triangle with a true parallel line; a real application needs positive segments and a closed geometry.
• • AD / DB = AE / EC is silent about DE itself. To get DE you also need BC and the similarity ratio AD / AB.

If the result looks off, check the pairing of D on AB with E on AC first, then the unit across the three inputs.

According to Wolfram MathWorld, two triangles are similar when their corresponding angles are equal and their sides are in the same ratio, the foundation for the AA similarity proof.

According to Khan Academy, AA similarity says two triangles are similar when two pairs of corresponding angles are equal, which makes AB / AD = AC / AE hold for any line parallel to BC.

When the triangle proportionality theorem sits on top of a right triangle, the Pythagorean Triples Calculator confirms the 3-4-5, 5-12-13, or 8-15-17 family that the proportion has to be consistent with.