Intercept Theorem Calculator (Thales) with Converse

This calculator uses the intercept theorem, also known as Thales’s intercept theorem, the basic proportionality theorem or the side splitter theorem, to find a missing length when two intersecting lines are cut by two parallel lines. Its converse tells you whether two lines are parallel. Every answer comes with a worked solution: the ratio equation written as fractions, the substituted values, the cross multiplication and a concluding sentence.

The intercept theorem states: let lines BM and CN intersect at A. If line MN is parallel to line BC, the corresponding lengths of triangles AMN and ABC are proportional: AM/AB = AN/AC = MN/BC. Triangle AMN is a scaled copy of triangle ABC and all three ratios equal the same scale factor k. A word of caution on names: in English, “Thales’s theorem” usually means a different result, that an angle inscribed in a semicircle is a right angle. The proportionality result used here is best known as the intercept theorem.

Two figures are covered. In the nested figure, M lies on segment AB and N on segment AC, so the small triangle AMN sits inside ABC. In the crossed (hourglass or bowtie) figure, M and B lie on opposite sides of A, as do N and C, and the two triangles meet at their common vertex A. The ratios are written in exactly the same way; in the calculator, simply name the smaller triangle AMN.

To find a length, keep the two ratios that contain the unknown and three known lengths, then cross-multiply. For example, with AM = 3, AB = 5 and AC = 10: 3/5 = AN/10, so AN = 3 × 10 ÷ 5 = 6. The calculator picks a complete ratio automatically, shows this working and rounds to two decimal places when the answer is not exact.

The converse proves that lines are parallel: if A, M, B and A, N, C are collinear in the same order and AM/AB = AN/AC, then MN is parallel to BC. Its contrapositive proves the opposite: if AM/AB ≠ AN/AC, the lines are not parallel. To compare two ratios without rounding, the calculator also checks the cross products AM × AC and AN × AB.

Conditions: the points must be collinear (stated in the problem, not assumed from a sketch), the lines must be given as parallel before you use the theorem, and the points must be in the same order on both lines before you use the converse. A complete written answer names the intersecting lines, the parallel lines and the theorem, writes the ratio equation, shows the calculation and ends with a sentence that includes the unit.