The right triangle altitude theorem or geometric mean theorem is a result in elementary geometry that describes a relation between the
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
on the
hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse e ...
in a
right triangle
A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
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
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved ...
''C'' and the diameter form a
right triangle
A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
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
In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is cons ...
), 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.
[*Hartmut Wellstein, Peter Kirsche: ''Elementargeometrie''. Springer, 2009, , pp. 76-77 (German, ) ]
Another application of provides a geometrical proof of the
AM–GM inequality
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; a ...
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 theorem can also be thought of as a special case of the
intersecting chords theorem for a circle, since the converse of
Thales' theorem
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved ...
ensures that the hypotenuse of the right angled triangle is the diameter of its
circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
.
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.
History
The theorem is usually attributed to
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
(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.
Proof
Based on similarity
Proof of theorem:
The triangles
and
are
similar, since:
* consider triangles
, here we have
and
, therefore by the
AA postulate
In Euclidean geometry, the AA postulate states that two triangles are similar if they have two corresponding angles congruent.
The AA postulate follows from the fact that the sum of the interior angles of a triangle is always equal to 180°. B ...
* 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.
Ilka Agricola
Ilka Agricola (born 8 August 1973 in The Hague)[Curriculum vitae](_blank)
retrieved 1 January 2017. , Thomas Friedrich: ''Elementary Geometry''. AMS 2008, , p. 25 ()
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 ''h
2'' 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 mapping
In plane geometry, a shear mapping is a linear map that displaces each point in a fixed direction, by an amount proportional to its signed distance from the line that is parallel to that direction and goes through the origin. This type of mappi ...
s (shear mappings preserve the area):
References
External links
''Geometric Mean''at
Cut-the-Knot
{{commonscat, Geometric mean theorem
Area
Articles containing proofs
Euclidean plane geometry
History of geometry
Theorems about right triangles