Geometric mean theorem
The right triangle altitude theorem or geometric mean theorem is a result in elementary geometry that describes a relation between the lengths of the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that the geometric mean of the two segments equals the altitude.
Theorem and applications
If h denotes the altitude in a right triangle and p and q the segments on the hypotenuse then the theorem can be stated as:
or in term of areas:
The latter version yields a method to square a rectangle with ruler and compass, that is to construct a square of equal area to a given rectangle. For such a rectangle with sides p and q we denote its top left vertex with D. Now we extend the segment q to its left by p (using arc AE centered on D) and draw a half circle with endpoints A and B with the new segment p+q as its diameter. Then we erect a perpendicular line to the diameter in D that intersects the half circle in C. Due to Thales' theorem C and the diameter form a right triangle with the line segment DC as its altitude, hence DC is the side of a square with the area of the rectangle. The method also allows for the construction of square roots (see constructible number), since starting with a rectangle that has a width of 1 the constructed square will have a side length that equals the square root of the rectangle's length.
The theorem can be used to provide a geometrical proof of the AM–GM inequality in the case of two numbers. For the numbers p and q one constructs a half circle with diameter p+q. Now the altitude represents the geometric mean and the radius the arithmetic mean of the two numbers. Since the altitude is always smaller or equal to the radius, this yields the inequality.
The geometric mean theorem can also be thought of as a special case of the intersecting chords theorem for a circle, since the converse of Thales' theorem ensures that the hypotenuse of the right angled triangle is the diameter of its circumcircle.
The converse statement is true as well. Any triangle, in which the altitude equals the geometric mean of the two line segments created by it, is a right triangle.
Historically the theorem is attributed to Euclid (ca. 360–280 BC), who stated it as a corollary to proposition 8 in book VI of his Elements. In proposition 14 of book II Euclid gives a method for squaring a rectangle, which essentially matches the method given here. Euclid however provides a different slightly more complicated proof for the correctness of the construction rather than relying on the geometric mean theorem.
Based on similarity
Proof of theorem:
The triangles and are similar, since:
- consider triangles , here we have and , therefore by the AA postulate
- further, consider triangles , here we have and , therefore by the AA postulate
Therefore, both triangles and are similar to and themselves, i.e. .
Because of the similarity we get the following equality of ratios and its algebraic rearrangement yields the theorem:
Proof of converse:
For the converse we have a triangle in which holds and need to show that the angle at C is a right angle. Now because of we also have . Together with the triangles and have an angle of equal size and have corresponding pairs of legs with the same ratio. This means the triangles are similar, which yields:
Based on the Pythagorean theorem
In the setting of the geometric mean theorem there are three right triangles , and , in which the Pythagorean theorem yields:
- , and
Adding the first 2 two equations and then using the third then leads to:
A division by two finally yields the formula of the geometric mean theorem.
Based on dissection and rearrangement
Dissecting the right triangle along its altitude h yields two similar triangles, which can be augmented and arranged in two alternative ways into a larger right triangle with perpendicular sides of lengths p+h and q+h. One such arrangement requires a square of area h2 to complete it, the other a rectangle of area pq. Since both arrangements yield the same triangle, the areas of the square and the rectangle must be identical.
Based on shear mappings
The square of the altitude can be transformed into an rectangle of equal area with sides p and q with the help of three shear mappings (shear mappings preserve the area):
- Claudi Alsina, Roger B. Nelsen: Icons of Mathematics: An Exploration of Twenty Key Images. MAA 2011, ISBN 9780883853528, pp. 31–32 (online copy, p. 31, at Google Books)
- Ilka Agricola, Thomas Friedrich: Elementary Geometry. AMS 2008, ISBN 9780821843475, p. 25 (online copy, p. 25, at Google Books)
- Hartmut Wellstein, Peter Kirsche: Elementargeometrie. Springer, 2009, ISBN 9783834808561, pp. 76-77 (German, online copy, p. 76, at Google Books)
- Euclid: Elements, book II – prop. 14, book VI – prop. 8, (online copy)
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