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Thales theorem calculator and converse
Calculate a missing length in a Thales configuration and get a complete worked solution. Converse mode compares ratios to determine whether the lines are parallel.
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A, M, B are assumed collinear in that order, and A, N, C collinear in that order.
The selected field is calculated automatically once three compatible lengths are available.
Enter the counterpart of the unknown and one complete corresponding pair. Small lengths must be shorter than their full lengths.
Enter AM, AB, AN and AC with AM < AB and AN < AC.
Worked solution
Why use this tool?
Use Thales’ theorem to calculate a missing length in a parallel-line configuration, or use its converse to check whether two lines are parallel.
Smart processing
Generate, convert, or validate technical data with purpose-built controls.
Supported formats
Technical formats and syntaxes suited to each generator, converter, or validator.
Save time
Get a clean, ready-to-use result in seconds without installing software or configuring a complex workflow.
Understanding the Thales formula
When M lies on [AB], N on [AC], and (MN) is parallel to (BC), corresponding lengths are proportional.
Thales theorem
AM / AB = AN / AC = MN / BC
Use two ratios with three known lengths to calculate the fourth.
Converse
AM / AB = AN / AC ⇒ (MN) ∥ (BC)
With points aligned in the same order, equal ratios establish parallelism.
- AM, AN, MN
- lengths of the smaller triangle AMN
- AB, AC, BC
- corresponding lengths of triangle ABC
Fonctionnement
How does this tool work?
The calculator applies proportional ratios between corresponding sides of triangles AMN and ABC when (MN) is parallel to (BC). In Theorem mode, it selects a fully known ratio, computes the scale factor, and uses it to derive the missing length. In Converse mode, it compares AM/AB with AN/AC using a small numerical tolerance and then generates a worked explanation that supports the conclusion about whether the two lines are parallel.
Use cases
Find a missing length
Calculate an unknown side when a valid Thales configuration and three compatible lengths are already known.
Check parallel lines
Use the converse to compare two length ratios and determine whether the corresponding lines can be considered parallel.
Write a worked proof
Generate the ratios, numerical substitution, and conclusion in a format that helps review or prepare a geometry exercise.
Guide
How to use this tool
- 1
Choose theorem or converse
The first mode calculates a length; the second tests parallelism.
- 2
Enter known lengths
Calculation runs automatically as soon as enough compatible values are available.
- 3
Read the worked solution
The tool shows the ratios, numeric substitution, and conclusion.
Tips and best practices
- Keep points in the A–M–B and A–N–C order for this configuration.
- Lengths of the smaller triangle must be shorter than the corresponding full lengths.
- Make sure each pair of compared sides occupies the same relative position in both triangles before writing the ratios.
Frequently asked questions
What is the difference between theorem and converse?
The theorem uses parallelism to calculate a length. The converse uses equal ratios to conclude that lines are parallel.
Is calculation automatic?
Yes. Results and the worked solution update as soon as the required values are valid.
Are calculations sent to a server?
No. The Thales engine runs directly in your browser.
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