Altitude (triangle)

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The three altitudes of a triangle intersect at the orthocenter, which for an acute triangle is inside the triangle.

In

orthogonal projection
.

Altitudes can be used in the computation of the area of a triangle: one-half of the product of an altitude's length and its base's length equals the triangle's area. Thus, the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the trigonometric functions.

In an

angle bisector
of the vertex angle.

It is common to mark the altitude with the letter h (as in height), often subscripted with the name of the side the altitude is drawn to.

Pythagoras' theorem
on the 3 triangles of sides (p + q, r, s ), (r, p, h ) and (s, h, q ),

In a right triangle, the altitude drawn to the hypotenuse c divides the hypotenuse into two segments of lengths p and q. If we denote the length of the altitude by hc, we then have the relation

  (Geometric mean theorem; see Special Cases, inverse Pythagorean theorem)
In a right triangle, the altitude from each acute angle coincides with a leg and intersects the opposite side at (has its foot at) the right-angled vertex, which is the orthocenter.

For acute triangles, the feet of the altitudes all fall on the triangle's sides (not extended). In an obtuse triangle (one with an

obtuse angle), the foot of the altitude to the obtuse-angled vertex falls in the interior of the opposite side, but the feet of the altitudes to the acute-angled vertices fall on the opposite extended side
, exterior to the triangle. This is illustrated in the adjacent diagram: in this obtuse triangle, an altitude dropped perpendicularly from the top vertex, which has an acute angle, intersects the extended horizontal side outside the triangle.

Orthocenter

Three altitudes intersecting at the orthocenter

The three (possibly extended) altitudes intersect in a single point, called the orthocenter of the triangle, usually denoted by H.[1][2] The orthocenter lies inside the triangle if and only if the triangle is acute. If one angle is a right angle, the orthocenter coincides with the vertex at the right angle.[2]

Let A, B, C denote the vertices and also the angles of the triangle, and let be the side lengths. The orthocenter has trilinear coordinates[3]

and

barycentric coordinates

Since barycentric coordinates are all positive for a point in a triangle's interior but at least one is negative for a point in the exterior, and two of the barycentric coordinates are zero for a vertex point, the barycentric coordinates given for the orthocenter show that the orthocenter is in an

obtuse triangle
.

In the complex plane, let the points A, B, C represent the numbers zA, zB, zC and assume that the circumcenter of triangle ABC is located at the origin of the plane. Then, the complex number

is represented by the point H, namely the altitude of triangle ABC.[4] From this, the following characterizations of the orthocenter H by means of free vectors can be established straightforwardly:

The first of the previous vector identities is also known as the problem of Sylvester, proposed by James Joseph Sylvester.[5]

Properties

Let D, E, F denote the feet of the altitudes from A, B, C respectively. Then:

  • The product of the lengths of the segments that the orthocenter divides an altitude into is the same for all three altitudes:[6][7]
The circle centered at H having radius the square root of this constant is the triangle's polar circle.[8]
  • The sum of the ratios on the three altitudes of the distance of the orthocenter from the base to the length of the altitude is 1:[9] (This property and the next one are applications of a more general property of any interior point and the three cevians through it.)
  • The sum of the ratios on the three altitudes of the distance of the orthocenter from the vertex to the length of the altitude is 2:[9]

Relation with circles and conics

Denote the

circumradius of the triangle by R. Then[12][13]

In addition, denoting r as the radius of the triangle's

excircles, and R again as the radius of its circumcircle, the following relations hold regarding the distances of the orthocenter from the vertices:[14]

If any altitude, for example, AD, is extended to intersect the circumcircle at P, so that AD is a chord of the circumcircle, then the foot D bisects segment HP:[7]

The

directrices of all parabolas that are externally tangent to one side of a triangle and tangent to the extensions of the other sides pass through the orthocenter.[15]

A

Relation to other centers, the nine-point circle

The orthocenter H, the

circumcenter O, and the center N of the nine-point circle all lie on a single line, known as the Euler line.[17] The center of the nine-point circle lies at the midpoint of the Euler line, between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half of that between the centroid and the orthocenter:[18]

The orthocenter is closer to the incenter I than it is to the centroid, and the orthocenter is farther than the incenter is from the centroid:

In terms of the sides a, b, c,

circumradius R,[19][20]
: p. 449 

Orthic triangle

Triangle abc (respectively, DEF in the text) is the orthic triangle of triangle ABC

If the triangle ABC is oblique (does not contain a right-angle), the pedal triangle of the orthocenter of the original triangle is called the orthic triangle or altitude triangle. That is, the feet of the altitudes of an oblique triangle form the orthic triangle, DEF. Also, the incenter (the center of the inscribed circle) of the orthic triangle DEF is the orthocenter of the original triangle ABC.[21]

Trilinear coordinates for the vertices of the orthic triangle are given by

The

collinear points.[22][23][21]

In any

acute triangle, the inscribed triangle with the smallest perimeter is the orthic triangle.[24] This is the solution to Fagnano's problem, posed in 1775.[25] The sides of the orthic triangle are parallel to the tangents to the circumcircle at the original triangle's vertices.[26]

The orthic triangle of an acute triangle gives a triangular light route.[27]

The tangent lines of the nine-point circle at the midpoints of the sides of ABC are parallel to the sides of the orthic triangle, forming a triangle similar to the orthic triangle.[28]

The orthic triangle is closely related to the tangential triangle, constructed as follows: let LA be the line tangent to the circumcircle of triangle ABC at vertex A, and define LB, LC analogously. Let The tangential triangle is A"B"C", whose sides are the tangents to triangle ABC's circumcircle at its vertices; it is

center of similitude of the orthic and tangential triangles, are on the Euler line.[20]
: p. 447 

Trilinear coordinates for the vertices of the tangential triangle are given by

The reference triangle and its orthic triangle are
orthologic triangles.

For more information on the orthic triangle, see here.

Some additional altitude theorems

Altitude in terms of the sides

For any triangle with sides a, b, c and semiperimeter the altitude from side a (the base) is given by

This follows from combining Heron's formula for the area of a triangle in terms of the sides with the area formula where the base is taken as side a and the height is the altitude from the vertex A (opposite side a).

By exchanging a with b or c, this equation can also used to find the altitudes hb and hc, respectively.

Inradius theorems

Consider an arbitrary triangle with sides a, b, c and with corresponding altitudes ha, hb, hc. The altitudes and the

incircle radius r are related by[29]
: Lemma 1 

Circumradius theorem

Denoting the altitude from one side of a triangle as ha, the other two sides as b and c, and the triangle's

circumradius (radius of the triangle's circumscribed circle) as R, the altitude is given by[30]

Interior point

If p1, p2, p3 are the perpendicular distances from any point P to the sides, and h1, h2, h3 are the altitudes to the respective sides, then[31]

Area theorem

Denoting the altitudes of any triangle from sides a, b, c respectively as ha, hb, hc, and denoting the semi-sum of the reciprocals of the altitudes as we have[32]

General point on an altitude

If E is any point on an altitude AD of any triangle ABC, then[33]: 77–78 

Triangle inequality

Since the area of the triangle is , the triangle inequality implies[34]

.

Special cases

Equilateral triangle

From any point P within an equilateral triangle, the sum of the perpendiculars to the three sides is equal to the altitude of the triangle. This is Viviani's theorem.

Right triangle

Comparison of the inverse Pythagorean theorem with the Pythagorean theorem

In a right triangle with legs a and b and hypotenuse c, each of the legs is also an altitude: and . The third altitude can be found by the relation[35][36]

This is also known as the inverse Pythagorean theorem.

Note in particular:

History

The theorem that the three altitudes of a triangle concur (at the orthocenter) is not directly stated in surviving

al-Nasawi in his (11th century) commentary on the Book of Lemmas, and attributed to al-Quhi (fl. 10th century).[38]

This proof in Arabic was translated as part of the (early 17th century) Latin editions of the Book of Lemmas, but was not widely known in Europe, and the theorem was therefore proven several more times in the 17th–19th century. Samuel Marolois proved it in his Geometrie (1619), and Isaac Newton proved it in an unfinished treatise Geometry of Curved Lines (c. 1680).[37] Later William Chapple proved it in 1749.[39]

A particularly elegant proof is due to

perpendicular bisectors of the new triangle, and therefore concur (at the circumcenter of the new triangle).[40]

See also

Notes

  1. ^ Smart 1998, p. 156
  2. ^ a b Berele & Goldman 2001, p. 118
  3. ^ Clark Kimberling's Encyclopedia of Triangle Centers "Encyclopedia of Triangle Centers". Archived from the original on 2012-04-19. Retrieved 2012-04-19.
  4. , page 90, Proposition 3
  5. , page 142
  6. ^ Johnson 2007, p. 163, Section 255
  7. ^ a b ""Orthocenter of a triangle"". Archived from the original on 2012-07-05. Retrieved 2012-05-04.
  8. ^ Johnson 2007, p. 176, Section 278
  9. ^ a b Panapoi, Ronnachai, "Some properties of the orthocenter of a triangle", University of Georgia.
  10. ^ Smart 1998, p. 182
  11. ^ Weisstein, Eric W. "Isotomic conjugate" From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/IsotomicConjugate.html
  12. ^ Weisstein, Eric W. "Orthocenter." From MathWorld--A Wolfram Web Resource.
  13. ^ Altshiller-Court 2007, p. 102
  14. ^ Bell, Amy, "Hansen's right triangle theorem, its converse and a generalization", Forum Geometricorum 6, 2006, 335–342.
  15. ^ Weisstein, Eric W. "Kiepert Parabola." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/KiepertParabola.html
  16. ^ Weisstein, Eric W. "Jerabek Hyperbola." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/JerabekHyperbola.html
  17. ^ Berele & Goldman 2001, p. 123
  18. ^ Berele & Goldman 2001, pp. 124-126
  19. ^ Marie-Nicole Gras, "Distances between the circumcenter of the extouch triangle and the classical centers", Forum Geometricorum 14 (2014), 51-61. http://forumgeom.fau.edu/FG2014volume14/FG201405index.html
  20. ^ a b Smith, Geoff, and Leversha, Gerry, "Euler and triangle geometry", Mathematical Gazette 91, November 2007, 436–452.
  21. ^ a b William H. Barker, Roger Howe (2007). "§ VI.2: The classical coincidences". Continuous symmetry: from Euclid to Klein. American Mathematical Society. p. 292. . See also: Corollary 5.5, p. 318.
  22. ^ Johnson 2007, p. 199, Section 315
  23. ^ Altshiller-Court 2007, p. 165
  24. ^ Johnson 2007, p. 168, Section 264
  25. ^ Berele & Goldman 2001, pp. 120-122
  26. ^ Johnson 2007, p. 172, Section 270c
  27. ^ Bryant, V., and Bradley, H., "Triangular Light Routes," Mathematical Gazette 82, July 1998, 298-299.
  28. ^ Dorin Andrica and Dan S ̧tefan Marinescu. "New Interpolation Inequalities to Euler's R ≥ 2r". Forum Geometricorum, Volume 17 (2017), pp. 149–156. http://forumgeom.fau.edu/FG2017volume17/FG201719.pdf
  29. ^ Johnson 2007, p. 71, Section 101a
  30. ^ Johnson 2007, p. 74, Section 103c
  31. ^ Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle," Mathematical Gazette 89, November 2005, 494.
  32. ^ Alfred S. Posamentier and Charles T. Salkind, Challenging Problems in Geometry, Dover Publishing Co., second revised edition, 1996.
  33. ^ Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle", Mathematical Gazette 89 (November 2005), 494.
  34. ^ Voles, Roger, "Integer solutions of ," Mathematical Gazette 83, July 1999, 269–271.
  35. ^ Richinick, Jennifer, "The upside-down Pythagorean Theorem," Mathematical Gazette 92, July 2008, 313–317.
  36. ^ a b Newton, Isaac (1971). "3.1 The 'Geometry of Curved Lines'". In Whiteside, Derek Thomas (ed.). The Mathematical Papers of Isaac Newton. Vol. 4. Cambridge University Press. pp. 454–455. Note Whiteside's footnotes 90–92, pp. 454–456.
  37. .
    Hogendijk, Jan P. (2008).
    "Two beautiful geometrical theorems by Abū Sahl Kūhī in a 17th century Dutch translation". Tārīk͟h-e ʾElm: Iranian Journal for the History of Science. 6: 1–36.
  38. . Cut The Knot. Retrieved 2019-11-17.
  39. ^
    Servois, Francois-Joseph
    (1804). Solutions peu connues de différens problèmes de Géométrie-pratique [Little-known solutions of various Geometry practice problems] (in French). Devilly, Metz et Courcier. p. 15.

    Gauss, Carl Friedrich (1810). "Zusätze". Geometrie der Stellung. By Carnot, Lazare (in German). Translated by Schumacher. republished in Gauss, Carl Friedrich (1873). "Zusätze". Werke. Vol. 4. Göttingen Academy of Sciences. p. 396.
    See
    .

References

External links