Intercept theorem

The intercept theorem, also known as Thales' theorem (not to be confused with another theorem with that name) or basic proportionality theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two intersecting lines are intercepted by a pair of parallels. It is equivalent to the theorem about ratios in similar triangles. Traditionally it is attributed to Greek mathematician Thales.Construction of a productConstruction of an inverseTo divide an arbitrary line segment A B ¯ {displaystyle {overline {AB}}} in a m : n {displaystyle m:n} ratio, draw an arbitrary angle in A with A B ¯ {displaystyle {overline {AB}}} as one leg. On the other leg construct m + n {displaystyle m+n} equidistant points, then draw the line through the last point and B and parallel line through the mth point. This parallel line divides A B ¯ {displaystyle {overline {AB}}} in the desired ratio. The graphic to the right shows the partition of a line segment A B ¯ {displaystyle {overline {AB}}} in a 5 : 3 {displaystyle 5:3} ratio.The intercept theorem can be used to determine a distance that cannot be measured directly, such as the width of a river or a lake, the height of tall buildings or similar. The graphic to the right illustrates measuring the width of a river. The segments | C F | {displaystyle |CF|} , | C A | {displaystyle |CA|} , | F E | {displaystyle |FE|} are measured and used to compute the wanted distance | A B | = | A C | | F E | | F C | {displaystyle |AB|={frac {|AC||FE|}{|FC|}}} .If the midpoints of two triangle sides are connected then the resulting line segment is parallel to the third triangle side.If the midpoints of the two non-parallel sides of a trapezoid are connected, then the resulting line segment is parallel to the other two sides of the trapezoid.Since C A ∥ B D {displaystyle CAparallel BD} , the altitudes of △ C D A {displaystyle riangle CDA} and △ C B A {displaystyle riangle CBA} are of equal length. As those triangles share the same baseline, their areas are identical. So we have | △ C D A | = | △ C B A | {displaystyle | riangle CDA|=| riangle CBA|} and therefore | △ S C B | = | △ S D A | {displaystyle | riangle SCB|=| riangle SDA|} as well. This yieldsDraw an additional parallel to S D {displaystyle SD} through A. This parallel intersects B D {displaystyle BD} in G. Then one has | A C | = | D G | {displaystyle |AC|=|DG|} and due to claim 1 | S A | | S B | = | D G | | B D | {displaystyle {frac {|SA|}{|SB|}}={frac {|DG|}{|BD|}}} and therefore | S A | | S B | = | A C | | B D | {displaystyle {frac {|SA|}{|SB|}}={frac {|AC|}{|BD|}}} Assume A C {displaystyle AC} and B D {displaystyle BD} are not parallel. Then the parallel line to A C {displaystyle AC} through D {displaystyle D} intersects S A {displaystyle SA} in B 0 ≠ B {displaystyle B_{0} eq B} . Since | S B | : | S A | = | S D | : | S C | {displaystyle |SB|:|SA|=|SD|:|SC|} is true, we have | S B | = | S D | | S A | | S C | {displaystyle |SB|={frac {|SD||SA|}{|SC|}}} and on the other hand from claim 2 we have | S B 0 | = | S D | | S A | | S C | {displaystyle |SB_{0}|={frac {|SD||SA|}{|SC|}}} . So B {displaystyle B} and B 0 {displaystyle B_{0}} are on the same side of S {displaystyle S} and have the same distance to S {displaystyle S} , which means B = B 0 {displaystyle B=B_{0}} . This is a contradiction, so the assumption could not have been true, which means A C {displaystyle AC} and B D {displaystyle BD} are indeed parallel ◻ {displaystyle square } The intercept theorem, also known as Thales' theorem (not to be confused with another theorem with that name) or basic proportionality theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two intersecting lines are intercepted by a pair of parallels. It is equivalent to the theorem about ratios in similar triangles. Traditionally it is attributed to Greek mathematician Thales.