Intercept theorem

The intercept theorem states that parallel lines cutting two transversals divide them into corresponding proportional segments. In a triangle, a line parallel to one side divides the other two sides proportionally.

Triangle proportionality theorem: A theorem stating that a line parallel to one side of a triangle divides the other two sides proportionally. This is the intercept theorem's familiar triangle form.
Indirect measurement: Determining an unknown length or distance from measurements and a mathematical model. The theorem can find a height or distance by comparing it with a measurable segment.
Euclidean geometry: The geometry of points, lines, angles, and figures governed by Euclid's parallel postulate. The standard intercept theorem is a result within Euclidean geometry.
Thales' intercept theorem: A theorem stating that a line parallel to one side of a triangle divides the other two sides proportionally. This name distinguishes the proportionality result from the right-angle theorem also called Thales' theorem.
Similar triangles: Triangles with equal corresponding angles and proportional corresponding side lengths. A parallel cut creates a smaller triangle similar to the original, yielding the segment ratios.
Surveying: The measurement and mapping of land, boundaries, and locations. Surveyors use proportional geometry to infer distances across obstacles.
Angle-angle similarity: A criterion stating that two triangles are similar when two pairs of corresponding angles are equal. Parallel lines supply equal angles, so this criterion proves the triangle form.
Thales' theorem: A theorem stating that an angle inscribed in a semicircle is a right angle. It is a different result that shares a traditional name with the intercept theorem.
Transversal: A line that intersects two or more other lines at distinct points. The theorem compares the segments formed when parallel lines meet transversals.
Shadow measurement: Estimating an object's height by comparing its shadow with the shadow of an object of known height. Parallel sunlight creates similar triangles, applying the same proportional reasoning.

Linked from 3 pages

Midpoint theoremRelated: It explains why bisecting two sides forces the joining segment to be parallel to the third.