Parallel Lines and Triangles Calculator

The Triangle Proportionality Theorem (also called the Side-Splitter Theorem or Basic Proportionality Theorem) states: if a line is drawn parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally. Symbolically: in triangle ABC with line DE drawn parallel to BC (D on AB, E on AC):

AD / DB = AE / EC

This tutorial covers the theorem, its converse, two worked examples, and how it underlies many triangle similarity proofs.

The setup

Take any triangle ABC. Draw a line DE inside the triangle such that:

• D lies on side AB
• E lies on side AC
• DE is parallel to side BC

The line DE "splits" sides AB and AC into two parts each. The theorem says these parts are in the same ratio.

The proportion

AD : DB = AE : EC

Equivalently: AD × EC = DB × AE (cross-multiplication form).

Also: AD / AB = AE / AC (proportions of the upper parts to the wholes).

Why the theorem is true

The parallel line creates similar triangles. △ADE ~ △ABC by AA:

• They share angle A.
• ∠ADE = ∠ABC (corresponding angles, DE ∥ BC).

By similarity, AD/AB = AE/AC = DE/BC. From AD/AB = AE/AC, basic algebra gives AD/DB = AE/EC (subtract 1 from each ratio: (AD−AB)/AB = (AE−AC)/AC, manipulate to AD/DB = AE/EC).

Worked example 1 — find unknown segment

Triangle ABC has line DE ∥ BC, with D on AB and E on AC. Given AD = 6, DB = 4, AE = 9. Find EC.

By the theorem: AD/DB = AE/EC → 6/4 = 9/EC → EC = (4 × 9) / 6 = 6.

Worked example 2 — find a missing segment elsewhere

Triangle XYZ has line PQ ∥ YZ, with P on XY and Q on XZ. Given XP = 5, PY = 3, XQ = 10. Find QZ.

XP/PY = XQ/QZ → 5/3 = 10/QZ → QZ = (3 × 10) / 5 = 6.

The converse

The converse holds too: if a line divides two sides of a triangle proportionally, then it is parallel to the third side.

So if AD/DB = AE/EC, then DE ∥ BC.

This converse is useful for proving lines parallel from segment-length data — a common task in geometry proofs.

The intercept theorem (multi-line extension)

The same proportionality works for any number of parallel lines crossing two transversals. Three parallel lines cut by two transversals produce proportional segments on both transversals — even outside the triangle context.

This is called the Intercept Theorem (or Thales' theorem in some textbooks). The Triangle Proportionality Theorem is the special case where one transversal becomes side AB and the other becomes side AC, with two parallel lines being BC and DE.

Worked example 3 — extension via converse

In triangle ABC, point D on AB has AD = 4 and DB = 6. Point E on AC has AE = 6 and EC = 9. Is line DE parallel to BC?

Check: AD/DB = 4/6 = 2/3. AE/EC = 6/9 = 2/3.

Ratios are equal, so by the converse, DE IS parallel to BC.

Real-world applications

• Surveying: measuring an inaccessible distance by setting up similar triangles with measurable proportional segments.
• Shadow methods: measuring tree heights or building heights via sun-shadow similar triangles.
• Engineering: scale models and proportional reasoning in design.
• Geometry proofs: the theorem is a building block for similarity theorems and many compass-straightedge constructions.

Common mistakes

• Confusing AD/DB with AD/AB. Theorem says "ratio of the two parts" (AD/DB), not "ratio of upper part to whole" (AD/AB). Both work in similar proportions but are different equations.
• Applying the theorem without parallel line. The proportionality only holds when DE ∥ BC. Without parallelism, segments do not divide proportionally.
• Inverting one of the ratios. AD/DB = AE/EC — both fractions in the same direction. Inverting just one (e.g., AD/DB = EC/AE) gives a different (wrong) relationship.