In Euclidean geometry, the **right triangle altitude theorem** or **geometric mean theorem** is a relation between 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.

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] }

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.^{ [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.

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 of theorem**:

The triangles △*ADC* , △* BCD* are similar, since:

- consider triangles △
*ABC*, △*ACD*; here we havetherefore by the AA postulate - further, consider triangles △
*ABC*, △*BCD*; here we havetherefore by the AA postulate

Therefore, both triangles △*ACD*, △*BCD* are similar to △*ABC* and themselves, i.e.

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

**Proof of converse:**

For the converse we have a triangle △*ABC* 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 △*ADC*, △*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:

In the setting of the geometric mean theorem there are three right triangles △*ABC*, △*ADC* and △*DBC* in which the Pythagorean theorem yields:

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

which finally yields the formula of the geometric mean theorem.^{ [4] }

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.

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):

In geometry a **quadrilateral** is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words *quadri*, a variant of four, and *latus*, meaning "side". It is also called a **tetragon**, derived from greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons. Since "gon" means "angle", it is analogously called a **quadrangle**, or 4-angle. 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 if 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**, formerly called a **rectangled triangle**, is a triangle in which one angle is a right angle, i.e., in which two sides are perpendicular. The relation between the sides and other angles of the right triangle is the basis for trigonometry.

In trigonometry, the **law of sines**, **sine law**, **sine formula**, or **sine rule** is an equation relating the lengths of the sides of any 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; its generalized version can be used with non-cyclic quadrilateral.

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.

In geometry, **symmedians** are three particular lines associated with every triangle. They are constructed by taking a median of the triangle, and reflecting the line over the corresponding angle bisector. The angle formed by the **symmedian** and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector.

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 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.

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 Euclidean geometry, a **bicentric quadrilateral** is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called *inradius* and *circumradius*, and *incenter* and *circumcenter* respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are **chord-tangent quadrilateral** and **inscribed and circumscribed quadrilateral**. It has also rarely been called a *double circle quadrilateral* and *double scribed quadrilateral*.

In mathematics, the **Pythagorean theorem** or **Pythagoras' theorem** is a fundamental relation in Euclidean geometry between 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 the hypotenuse *c*, often called the **Pythagorean equation**:

An **integer triangle** or **integral triangle** is a triangle all of whose side lengths are integers. A **rational triangle** is one whose side lengths are rational numbers; any rational triangle can be rescaled by the lowest common denominator of the sides to obtain a similar integer triangle, so there is a close relationship between integer triangles and rational triangles.

In Euclidean geometry, an **orthodiagonal quadrilateral** is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

**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.

- 1 2 3 4 5
- Hartmut Wellstein, Peter Kirsche:
*Elementargeometrie*. Springer, 2009, ISBN 9783834808561, pp. 76-77 (German,*online copy*, p. 76, at Google Books)

- Hartmut Wellstein, Peter Kirsche:
- ↑ 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) - ↑ Euclid:
*Elements*, book II – prop. 14, book VI – pro6767800hshockedmake ,me uoppppp. 8, (online copy) - ↑ Ilka Agricola, Thomas Friedrich:
*Elementary Geometry*. AMS 2008, ISBN 9780821843475, p. 25 (*online copy*, p. 25, at Google Books)

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