The intercept theorem in space: hypotheses and a converse

Content updated: October 22, 2026

Parallel planes determine the same division ratio on transversal lines. For a converse, retain point order or use directed ratios: length ratios alone can lose positional information.

On this page: Background · Method · Examples · Practice · Further study

1. The theorem and a conditional converse

Each line must meet all three planes at finite points and therefore is not parallel to them. The points in each triple are distinct because the planes are distinct. These are length ratios, with denominators different from $0$.

Reduce to the plane theorem as follows. Through $A_2$, draw $d^{\prime}$ with the same direction as the first line, meeting the remaining planes at $B^{\prime},C^{\prime}$. If $d^{\prime}$ differs from the first line, parallelograms in the plane of those parallel lines give $A_2B^{\prime}=A_1B_1$ and $B^{\prime} C^{\prime}=B_1C_1$. If they coincide, these equalities hold directly. If $d^{\prime}$ differs from the second line, they meet at $A_2$; in their plane, the three parallel planes cut three parallel lines. The plane intercept theorem yields the required ratio. If the lines coincide, the ratios agree directly.

The converse used in this lesson: let two lines be skew, with distinct points $A_i,B_i,C_i$ on line $i$, and with $B_i$ between $A_i,C_i$ in both triples. If the two ratios above agree, the lines $A_1A_2,B_1B_2,C_1C_2$ are all parallel to some plane. Infinitely many such planes exist.

2. Why does point order matter?

The common ratio and order give $t\in(0,1)$ such that $B_i=(1-t)A_i+tC_i$ in coordinates or position-vector notation. Hence

The two vectors on the right are not parallel: otherwise their four endpoints would be coplanar, contradicting the original skew lines. They span the directions of a plane, including direction $B_1B_2$. Choose a plane with those directions and translate it so that it contains none of the three connecting lines. It is then parallel to all three. In this family, at most three positions contain a connecting line, leaving infinitely many other choices.

A length ratio without point order permits interior and exterior division in different ways. One cannot then infer the same parameter $t$. A statement covering all orders must use the corresponding directed ratios.

3. Worked examples

Example 1. A moving line parallel to a fixed plane

Let $ABCD$ be a nondegenerate tetrahedron. Points $M,N$ lie inside the edges $BC,AD$, with $BM/MC=AN/ND$. Prove that $MN$ is always parallel to a fixed plane.

Choose $E$ in $(BCD)$ so that $BCDE$ is a parallelogram in that vertex order. Then $BE\parallel CD$, and $(\alpha)=(ABE)$ contains the directions $AB,CD$. The plane is fixed because the tetrahedron vertices are fixed.

The ratio and interior positions give $t\in(0,1)$ such that $M=(1-t)B+tC$ and $N=(1-t)A+tD$. Thus

This vector has direction parallel to $(ABE)$. For disjoint parallelism, also establish $M\notin(ABE)$. Indeed, $C\notin(ABE)$: if $C$ belonged to that plane, $E=B+D-C$ would place $D$ there as well, making the tetrahedron degenerate. Since $M$ is interior to $BC$ and $B\in(ABE)$, we have $M\notin(ABE)$. Therefore $MN\parallel(ABE)$.

If $t=0$ is allowed, $M=B,N=A$ and $MN=AB\subset(ABE)$; disjoint parallelism no longer follows. Thus the interior-edge condition is necessary for the stated conclusion.

Example 2. Segment lengths from the intercept theorem

Three parallel planes meet two lines in the orders $A_1,B_1,C_1$ and $A_2,B_2,C_2$. Given $A_1B_1=6$, $B_1C_1=4$, and $A_2B_2=9$, we have $6/4=9/B_2C_2$, so $B_2C_2=6$.

4. Practice

1. For three parallel planes as in the theorem, given $A_1B_1=3$, $B_1C_1=2$, and $B_2C_2=4$, find $A_2B_2$.
2. In Example 1, if $BM/MC=2$, what is the parameter $t=BM/BC$? Express $\overrightarrow{MN}$ in terms of $\overrightarrow{BA},\overrightarrow{CD}$.
3. Why does a direction parallel to a plane not itself guarantee that a line has no common point with the plane?

5. Common mistakes

• Omitting finite or distinct intersections, or allowing a denominator $0$.
• Using length ratios for the converse without preserving point order.
• Confusing a fixed orientation with a fixed plane position; in the example, explain that $(ABE)$ is constructed from fixed points.
• Claiming parallelism when the line actually lies in the plane.

6. Related lessons and resources

• Lines parallel to planes (English lesson)
• Parallel planes (English lesson)
• Cross-sections parallel to a plane (English lesson)

Open the original practice resource on Scribd. The practice problems in this lesson are ready to use.

Images linked from the old resource for reference: Figure 1 · Figure 2.