What does this tool do?
This calculator applies Thales’s theorem, also called the intercept theorem, to the classic figure: a triangle ABC, a point M on line (AB), a point N on line (AC) and a line (MN) parallel to (BC). Pick the length you are looking for among AM, AB, AN, AC, MN and BC, enter the others you know, and it gives the answer by cross multiplication.
In the second mode it uses the converse of Thales’s theorem: from AM, AB, AN and AC it compares the two ratios and tells you whether (MN) and (BC) are parallel. Both modes work for the triangle figure and for the butterfly figure, where the two triangles meet at A like an X.
Each result comes with the working laid out as in a Moroccan collège exercise: the theorem, the equal ratios with the numbers, the cross multiplication and the conclusion. When the figure is a right triangle, the Pythagorean theorem calculator gives the missing sides.
How to use it
- Choose “Find a length” or “Converse”, then the figure: triangle or butterfly.
- Pick the length you are looking for and enter the known lengths, in the same unit.
- Read the result and the step-by-step solution: equal ratios, cross multiplication and conclusion.
The calculator needs the partner of the unknown length and one complete pair. To find AN, for example, enter AC and either AM and AB, or MN and BC. If you fill in both pairs, their ratios must agree. Lengths can be decimals with a point or a comma, or fractions such as 3/4.
Calculation method
AN/AC = AM/AB ⇒ AN × AB = AC × AM ⇒ AN = AC × AM ÷ AB
AM/AB = AN/AC (same order of points) ⇒ (MN) ∥ (BC)
The three ratios are equal because triangle AMN is a reduction (or an enlargement) of triangle ABC. Each ratio puts a side of the small triangle over the matching side of the large one: AM over AB, AN over AC, MN over BC.
To find a length, write the equality of the two ratios that contain it and a known ratio, then cross-multiply. With AN/AC = AM/AB, you get AN × AB = AC × AM, so AN = AC × AM ÷ AB. The calculator keeps exact fractions, so 10/3 stays 10/3 and is also shown as about 3.3333.
For the converse, the two ratios AM/AB and AN/AC are computed separately and compared exactly. The cross products AM × AC and AN × AB give the same answer without dividing, which is a useful check.
Worked example
Example 1: find AN and MN
- In triangle ABC, M is on [AB], N is on [AC] and (MN) ∥ (BC). We know AM = 2 cm, AB = 5 cm, AC = 7.5 cm and BC = 6 cm.
- By Thales’s theorem: AM/AB = AN/AC = MN/BC, so 2/5 = AN/7.5 = MN/6.
- Cross multiplication: AN = 7.5 × 2 ÷ 5 = 15 ÷ 5 = 3.
- Same for MN: MN = 6 × 2 ÷ 5 = 12 ÷ 5 = 2.4.
AN = 3 cm and MN = 2.4 cm. The ratio is 2/5 = 0.4: triangle AMN is a reduction of ABC.
Another scenario: are (MN) and (BC) parallel?
- A, M, B and A, N, C are lined up in this order, with AM = 3, AB = 8, AN = 4.5 and AC = 12.
- First ratio: AM/AB = 3/8 = 0.375. Second ratio: AN/AC = 4.5/12 = 0.375.
- Check with the cross products: AM × AC = 3 × 12 = 36 and AN × AB = 4.5 × 8 = 36.
The ratios are equal and the points are in the same order, so by the converse of Thales’s theorem, (MN) ∥ (BC).
Understanding the results
The ratio k
The common value of the three ratios is the scale factor between the two triangles. With k = 0.4, every side of AMN is 0.4 times the matching side of ABC, and its area is k² = 0.16 times as large.
Triangle or butterfly
In the triangle figure, M lies between A and B, so AM is always shorter than AB, and the calculator checks it. In the butterfly figure, M and N are on the other side of A and k can be any positive number, even larger than 1.
Why the order of points matters
The converse needs A, M, B and A, N, C in the same order. Equal ratios with M on [AB] but N beyond A would describe a different figure where (MN) is not parallel to (BC). That is why you choose the figure before checking.
Measuring a height with a shadow
Thales is said to have measured a pyramid this way. A stick 1.5 m tall casts a shadow 2 m long; at the same moment a tree casts a shadow 12 m long. The sun’s rays are parallel, so the tree is 12 × 1.5 ÷ 2 = 9 m tall.
Tips and limitations
- Write the three ratios in the same order every time: small triangle on top, large triangle below.
- Use only the two ratios you need: the one with the unknown and one that is fully known.
- Check that your answer is shorter than its partner in the triangle figure: AN must be shorter than AC.
- A missing length is a rule of three in disguise: the cross multiplication is the same.
Mistakes to avoid
- Mixing up the ratios, for example AM/AB = AC/AN: the small triangle must always be on the same side.
- Using MB instead of AB: the ratio needs whole sides measured from A.
- Applying the theorem without parallel lines, or the converse without checking the order of the points.
- Concluding “parallel” from ratios that are only close because the lengths were rounded.
Frequently asked questions
What does Thales’s theorem say?
If M is on line (AB), N is on line (AC) and (MN) is parallel to (BC), then AM/AB = AN/AC = MN/BC. Triangles AMN and ABC have the same proportions. It is also called the intercept theorem.
How do I use the converse of Thales’s theorem?
Work out AM/AB and AN/AC separately. If they are equal and A, M, B and A, N, C are lined up in the same order, then (MN) and (BC) are parallel. If they are different, the lines are not parallel.
What is the butterfly figure?
It is the figure where M and N are on the other side of A, so that (BM) and (CN) cross at A and form an X. The theorem works the same way, but AM can be longer than AB.
Can I use Thales’s theorem with MB instead of AB?
Not directly. First work out AB = AM + MB in the triangle figure, then use AM/AB. The calculator asks for AB for this reason.
What if my two ratios are almost equal?
The calculator compares the exact fractions. If they differ by less than 1%, it says so: with measured lengths, the lines are probably parallel, but the exact calculation does not prove it.