Similar Triangles with Parallel Lines Calculator

The Basic Proportionality Theorem (BPT) — also called Thales' Theorem in some curricula — is one of the most useful similarity tools in plane geometry. It states: if a line drawn parallel to one side of a triangle intersects the other two sides at distinct points, then it divides those two sides in the same ratio. From BPT come the AA similarity setup, the Intercept Theorem (extension to multiple parallel lines), and a remarkable family of "indirect measurement" problems.

The theorem stated precisely

Consider triangle ABC. Let line DE be parallel to side BC, with D lying on AB and E lying on AC.

BPT: AD / DB = AE / EC.

The two sides AB and AC are divided by line DE in the SAME ratio. This is true regardless of where DE lies (as long as it's parallel to BC and intersects the other two sides at distinct points).

Why the theorem is true — similar triangles

Draw line DE parallel to BC. The smaller triangle ADE sits at the top of the larger triangle ABC. Because DE ∥ BC, the corresponding angles formed by lines AB and AC at the parallel cuts are equal:

• ∠ADE = ∠ABC (corresponding angles, DE ∥ BC, transversal AB)
• ∠AED = ∠ACB (corresponding angles, DE ∥ BC, transversal AC)

The two triangles share angle ∠A. So △ADE ~ △ABC by AA Similarity.

Similar triangles have proportional corresponding sides: AD/AB = AE/AC = DE/BC.

BPT follows: if AD/AB = AE/AC, then by subtraction AD/AB − AE/AC also relates DB/AB and EC/AC. Algebra gives AD/DB = AE/EC.

Worked example 1 — Find unknown segment via BPT

In △ABC, line DE ∥ BC with D on AB, E on AC. Given AD = 4, DB = 6, AE = 5. Find EC.

By BPT: AD / DB = AE / EC
4 / 6 = 5 / EC
EC = (6 × 5) / 4 = 7.5

So EC = 7.5. Total AC = AE + EC = 5 + 7.5 = 12.5.

Worked example 2 — Verify a line is parallel

In △ABC, line DE has D on AB with AD = 3, DB = 6, and E on AC with AE = 4, EC = 8.

Check the ratios: AD/DB = 3/6 = 0.5. AE/EC = 4/8 = 0.5. Equal — so by the converse of BPT, line DE IS parallel to BC.

The converse of BPT: if a line divides two sides of a triangle proportionally, the line is parallel to the third side. This is the standard tool for PROVING parallelism from segment lengths.

From BPT to AA similarity

BPT is the engine behind the most-used similarity proof pattern in geometry:

This 5-line proof is the standard answer to "prove these triangles are similar using parallel lines".

The Intercept Theorem (multi-line extension)

If THREE or more parallel lines are cut by two transversals, the segments they cut on the transversals are proportional. This generalizes BPT from a triangle (with one cut) to a set of parallel lines (with any number of cuts).

Formal statement: lines ℓ₁ ∥ ℓ₂ ∥ ℓ₃ cut by transversals t₁ and t₂. Let the cut points on t₁ be A, B, C (in order) and on t₂ be A', B', C' (in order). Then AB / BC = A'B' / B'C'.

Worked example 3 — Intercept Theorem

Three parallel lines are crossed by two transversals. The first transversal's segments measure 4 (upper) and 6 (lower). The second transversal's upper segment is 3. Find the lower segment.

By the Intercept Theorem: 4/6 = 3/x → x = (6 × 3) / 4 = 4.5.

BPT in coordinate geometry — the section formula

BPT shows up in coordinate geometry too. If point E divides segment AC in the ratio m:n, then BPT (applied with a parallel line through E) gives the section formula. The "section formula" is essentially BPT translated into x-y coordinates. See the Section Formula Calculator for the coordinate version.

BPT in real-world indirect measurement

The shadow-stick method for measuring a tree's height is a BPT problem. You and the tree both cast shadows in the same sunlight. The two triangles formed (you + your shadow + the sun ray; tree + tree shadow + sun ray) are similar by AA — and the sun rays are parallel (because the sun is essentially at infinity).

By BPT-style proportions: tree height / tree shadow = your height / your shadow. Measure all three, plug in, get the tree height.

Common mistakes

• Confusing which sides are divided. BPT divides the two sides ADJACENT to the original side. If DE is drawn parallel to BC, then DE divides AB and AC. Not BC.
• Writing the ratio incorrectly. The correct BPT ratio is "upper over lower for each side": AD/DB = AE/EC. Mixing them (AD/EC = DB/AE) is wrong.
• Applying BPT to non-parallel lines. The theorem requires DE ∥ BC. Without parallelism, the proportional division does not hold.
• Confusing BPT with congruence. BPT establishes SIMILARITY (proportional sides), not CONGRUENCE (equal sides). The smaller triangle ADE is similar to ABC but smaller in size.
• Forgetting the converse needs strict equality of ratios. If you want to prove a line is parallel via the converse of BPT, the two ratios must be EXACTLY equal — close-but-not-equal does not prove parallelism.