Intercept Theorem Calculator
This intercept theorem calculator fills in missing lengths: enter at least three known segments and the tool completes the rest. Example: ZA = 4, ZB = 10, ZC = 6 and AC = 3 give ZD = 15 and BD = 7.5.
Source: Wikipedia – Strahlensätze. Updated: .
How it is calculated
The intercept theorem describes how lengths relate when two rays start at one point Z and are cut by two parallel lines. Points A and B lie on the first ray, C and D on the second, and AC is parallel to BD. Then the ratios of corresponding lengths are all equal.
First and second intercept theorem
The first theorem compares pieces on the rays: ZA : ZB = ZC : ZD. The second includes the parallels: AC : BD = ZA : ZB. Together they give the scale factor k = ZB : ZA = ZD : ZC = BD : AC. The calculator first finds k from a pair where both lengths are known and then computes every missing length. The pieces AB and CD are the differences ZB − ZA and ZD − ZC; you can also enter them yourself when a problem only gives ZA and AB.
Example: height of a tree
A 1.80 m stick casts a 2.40 m shadow while the tree beside it casts a 12 m shadow. With ZA = 2.40, AC = 1.80 and ZB = 12 the calculator returns BD = 9 m as the tree height, because k = 12 : 2.40 = 5.
- A and C are always the points closer to the centre Z; keep the same order on both rays.
- If the ratios do not agree, the calculator tells you that AC and BD cannot be parallel.
- The steps at the end show both theorems with your numbers.
Frequently asked questions
How do you use the intercept theorem?
Find the ratio from a pair where both lengths are known, for example k = ZB : ZA. Then multiply or divide each unknown length by it: ZD = ZC · k and BD = AC · k.
What is the difference between the first and second intercept theorem?
The first compares lengths on the rays (ZA : ZB = ZC : ZD), the second compares the parallels with the ray sections (AC : BD = ZA : ZB). The scale factor is the same in both.
When can the intercept theorem be applied?
When two parallel lines are cut by two rays that start at the same point. The converse is useful to test whether two lines really are parallel.
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