# Geometric mean theorem

Last updated

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: [1]

${\displaystyle h={\sqrt {pq}}}$

or in term of areas:

${\displaystyle h^{2}=pq.}$

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. [1]

Another application of provides 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. [2]

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 ensures that the hypotenuse of the right angled triangle is the diameter of its circumcircle. [1]

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 (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. [1] [3]

## Proof

### Based on similarity

Proof of theorem:

The triangles ${\displaystyle \triangle ADC}$ and ${\displaystyle \triangle BCD}$ are similar, since:

• consider triangles ${\displaystyle \triangle ABC,\triangle ACD}$, here we have ${\displaystyle \angle ACB=\angle ADC=90^{\circ }}$ and ${\displaystyle \angle BAC=\angle CAD}$, therefore by the AA postulate ${\displaystyle \triangle ABC\sim \triangle ACD}$
• further, consider triangles ${\displaystyle \triangle ABC,\triangle BCD}$, here we have ${\displaystyle \angle ACB=\angle BDC=90^{\circ }}$ and ${\displaystyle \angle ABC=\angle CBD}$, therefore by the AA postulate ${\displaystyle \triangle ABC\sim \triangle BCD}$

Therefore, both triangles ${\displaystyle \triangle ACD}$ and ${\displaystyle \triangle BCD}$ are similar to ${\displaystyle \triangle ABC}$ and themselves, i.e. ${\displaystyle \triangle ACD\sim \triangle ABC\sim \triangle BCD}$.

Because of the similarity we get the following equality of ratios and its algebraic rearrangement yields the theorem:. [1]

${\displaystyle {\frac {h}{p}}={\frac {q}{h}}\,\Leftrightarrow \,h^{2}=pq\,\Leftrightarrow \,h={\sqrt {pq}}\qquad (h,p,q>0)}$

Proof of converse:

For the converse we have a triangle ${\displaystyle \triangle ABC}$ in which ${\displaystyle h^{2}=pq}$ holds and need to show that the angle at C is a right angle. Now because of ${\displaystyle h^{2}=pq}$ we also have ${\displaystyle {\tfrac {h}{p}}={\tfrac {q}{h}}}$. Together with ${\displaystyle \angle ADC=\angle CDB}$ the triangles ${\displaystyle \triangle ADC}$ and ${\displaystyle \triangle BDC}$ have an angle of equal size and have corresponding pairs of legs with the same ratio. This means the triangles are similar, which yields:

${\displaystyle \angle ACB=\angle ACD+\angle DCB=\angle ACD+(90^{\circ }-\angle DBC)=\angle ACD+(90^{\circ }-\angle ACD)=90^{\circ }}$

### Based on the Pythagorean theorem

In the setting of the geometric mean theorem there are three right triangles ${\displaystyle \triangle ABC}$, ${\displaystyle \triangle ADC}$ and ${\displaystyle \triangle DBC}$, in which the Pythagorean theorem yields:

${\displaystyle h^{2}=a^{2}-q^{2}}$, ${\displaystyle h^{2}=b^{2}-p^{2}}$ and ${\displaystyle c^{2}=a^{2}+b^{2}}$

Adding the first 2 two equations and then using the third then leads to:

${\displaystyle 2h^{2}=a^{2}+b^{2}-p^{2}-q^{2}=c^{2}-p^{2}-q^{2}=(p+q)^{2}-p^{2}-q^{2}=2pq}$.

A division by two finally yields the formula of the geometric mean theorem. [4]

### 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):

## Related Research Articles

A quadrilateral is a polygon in Euclidean plane geometry with four edges (sides) and four vertices (corners). Other names for quadrilateral include quadrangle, tetragon, and 4-gon. A quadrilateral with vertices , , and is sometimes denoted as .

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted .

In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling, possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other.

A right triangle or right-angled triangle (British), or more formally an orthogonal triangle, is a triangle in which one angle is a right angle. The relation between the sides and angles of the right angled is the basis for trigonometry.

In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects.

In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle to the sines of its angles. According to the law,

In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base. This line containing the opposite side is called the extended base of the altitude. The intersection of the extended base and the altitude is called the foot of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude at that vertex. It is a special case of orthogonal projection.

In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.

In Euclidean geometry, Brahmagupta's formula is used to find the area of any cyclic quadrilateral given the lengths of the sides.

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 equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 and the other has a length of 4, then their squares add up to 25. The length of the hypotenuse is the square root of 25, that is, 5.

In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC 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 as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras.

A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.

In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.

In a right triangle, a cathetus, commonly known as a leg, is either of the sides that are adjacent to the right angle. It is occasionally called a "side about the right angle". The side opposite the right angle is the hypotenuse. In the context of the hypotenuse, the catheti are sometimes referred to simply as "the other two sides".

In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. The line through these points is the Simson line of P, named for Robert Simson. The concept was first published, however, by William Wallace in 1799.

A Kepler triangle is a right triangle with edge lengths in geometric progression. The ratio of the progression is 𝜙, where 𝜙 is the golden ratio, and can be written: , or approximately 1 : 1.272 : 1.618. The squares of the edges of this triangle are also in geometric progression according to the golden ratio itself.

The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate.

In mathematics, the Pythagorean theorem, or Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:

Pappus's area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. The theorem, which can also be thought of as a generalization of the Pythagorean theorem, is named after the Greek mathematician Pappus of Alexandria, who discovered it.

## References

• Hartmut Wellstein, Peter Kirsche: Elementargeometrie. Springer, 2009, ISBN   9783834808561, pp. 76-77 (German, online copy , p. 76, at Google Books)
1. 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)
2. Euclid: Elements, book II – prop. 14, book VI – pro6767800hshockedmake ,me uoppppp. 8, (online copy)
3. Ilka Agricola, Thomas Friedrich: Elementary Geometry. AMS 2008, ISBN   9780821843475, p. 25 ( online copy , p. 25, at Google Books)