User:RDBury/Mean proportional

In mathematics, especially in Euclidean geometry, a mean proportional of two given quantities a and b is a quantity r so that a:r=r:b. Algebraically, this is the same as saying r²=ab or r is the geometric mean of a and b, but the mean proportional terminology is often used in historical contexts. For example 6 is a mean proportional between 2 and 18.

More generally, two or more mean proportionals for given quantities a and b can be defined as quantities which can be placed between a and b in a geometric sequence. For example r₁ and r₂ are two mean proportionals of a and b if a, r₁, r₂, b forms a geometric sequence, or equivalently, a:r₁=r₁:r₂=r₂:b. The problem of finding two mean proportionals is the equivalent in geometry to the problem of finding cube roots in algebra. Thus, the problem of Doubling the cube is equivalent that of finding two mean proportionals between the a line segment and a segment of twice the length.

Construction

Construction of a mean proportional

The construction of a mean proportion between two given line segments is the subject of Proposition 13 in Book VI of Euclid's Elements.

Euclid's construction is given as follows: Given lengths a and b, on a line mark off segments AH = a and HC = b. Find the midpoint of AC and using it as a center, draw a semicircle with diameter AC. Construct a perpendicular to the line at H and let B be the point where it intersects the semicircle. Then $\angle ABC$ is a right angle and $\angle ABH=\angle BCH$. This implies $\triangle ABH$ and $\triangle BCH$ are similar triangles and AH:HB = HB:HC, in other words HB is the mean proportion of a and b.