In elementary geometry, proportionality often involves parallel lines, through the intercept theorem. But there are also antiparallel lines. Might there likewise be such a thing as "antiproportionality"?
A celebrated geometry textbook written by Charles de Comberousse in 1861 states: "When two lines respectively make equal angles with the sides of the same angle, they are called antiparallel lines". The two figures below are then set side by side.
The first figure illustrates the intercept theorem. The textbook does not use the French name for this theorem, "Thales's theorem", as that term came into use slightly later. Elsewhere, it is often called the "intercept theorem" or the "proportionality theorem".
Lines (BC ) and (DE ) are parallel, and we obtain the proportion ADAB=AEAC or equivalently ACAB=AEAD because triangles ABC and ADE are similar (see box), with corresponding vertices D and B, on the one hand, and E and C, on the other.
