Altitude (triangle)
In
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
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.
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
For acute triangles, the feet of the altitudes all fall on the triangle's sides (not extended). In an obtuse triangle (one with an
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
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
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]
- The circumcenter of the triangle.[10]
- The isotomic conjugate of the orthocenter is the symmedian point of the anticomplementary triangle.[11]
- Four points in the plane, such that one of them is the orthocenter of the triangle formed by the other three, is called an orthocentric system or orthocentric quadrangle.
Relation with circles and conics
Denote the
In addition, denoting r as the radius of the triangle's
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
A
Relation to other centers, the nine-point circle
The orthocenter H, the
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,
: p. 449Orthic triangle
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
In any
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
Trilinear coordinates for the vertices of the tangential triangle are given by
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
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
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
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
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
See also
Notes
- ^ Smart 1998, p. 156
- ^ a b Berele & Goldman 2001, p. 118
- ^ Clark Kimberling's Encyclopedia of Triangle Centers "Encyclopedia of Triangle Centers". Archived from the original on 2012-04-19. Retrieved 2012-04-19.
- ISBN 978-0-8176-4326-3, page 90, Proposition 3
- ISBN 0-486-61348-8, page 142
- ^ Johnson 2007, p. 163, Section 255
- ^ a b ""Orthocenter of a triangle"". Archived from the original on 2012-07-05. Retrieved 2012-05-04.
- ^ Johnson 2007, p. 176, Section 278
- ^ a b Panapoi, Ronnachai, "Some properties of the orthocenter of a triangle", University of Georgia.
- ^ Smart 1998, p. 182
- ^ Weisstein, Eric W. "Isotomic conjugate" From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/IsotomicConjugate.html
- ^ Weisstein, Eric W. "Orthocenter." From MathWorld--A Wolfram Web Resource.
- ^ Altshiller-Court 2007, p. 102
- ^ Bell, Amy, "Hansen's right triangle theorem, its converse and a generalization", Forum Geometricorum 6, 2006, 335–342.
- ^ Weisstein, Eric W. "Kiepert Parabola." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/KiepertParabola.html
- ^ Weisstein, Eric W. "Jerabek Hyperbola." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/JerabekHyperbola.html
- ^ Berele & Goldman 2001, p. 123
- ^ Berele & Goldman 2001, pp. 124-126
- ^ 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
- ^ a b Smith, Geoff, and Leversha, Gerry, "Euler and triangle geometry", Mathematical Gazette 91, November 2007, 436–452.
- ^ a b
William H. Barker, Roger Howe (2007). "§ VI.2: The classical coincidences". Continuous symmetry: from Euclid to Klein. American Mathematical Society. p. 292. ISBN 978-0-8218-3900-3. See also: Corollary 5.5, p. 318.
- ^ Johnson 2007, p. 199, Section 315
- ^ Altshiller-Court 2007, p. 165
- ^ Johnson 2007, p. 168, Section 264
- ^ Berele & Goldman 2001, pp. 120-122
- ^ Johnson 2007, p. 172, Section 270c
- ^ Bryant, V., and Bradley, H., "Triangular Light Routes," Mathematical Gazette 82, July 1998, 298-299.
- ISBN 0-06-500006-4
- ^ 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
- ^ Johnson 2007, p. 71, Section 101a
- ^ Johnson 2007, p. 74, Section 103c
- ^ Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle," Mathematical Gazette 89, November 2005, 494.
- ^ Alfred S. Posamentier and Charles T. Salkind, Challenging Problems in Geometry, Dover Publishing Co., second revised edition, 1996.
- ^ Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle", Mathematical Gazette 89 (November 2005), 494.
- ^ Voles, Roger, "Integer solutions of ," Mathematical Gazette 83, July 1999, 269–271.
- ^ Richinick, Jennifer, "The upside-down Pythagorean Theorem," Mathematical Gazette 92, July 2008, 313–317.
- ^ 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.
- .
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. - doi:10.1080/14786445008646583. Footnote on pp. 207–208. Quoted by Bogomolny, Alexander (2010). "A Possibly First Proof of the Concurrence of Altitudes". Cut The Knot. Retrieved 2019-11-17.
- ^
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
- Altshiller-Court, Nathan (2007) [1952], College Geometry, Dover
- Berele, Allan; Goldman, Jerry (2001), Geometry: Theorems and Constructions, Prentice Hall, ISBN 0-13-087121-4
- Bogomolny, Alexander. "Existence of the Orthocenter". Cut the Knot. Retrieved 2022-12-17.
- Johnson, Roger A. (2007) [1960], Advanced Euclidean Geometry, Dover, ISBN 978-0-486-46237-0
- Smart, James R. (1998), Modern Geometries (5th ed.), Brooks/Cole, ISBN 0-534-35188-3
External links
- Weisstein, Eric W. "Altitude". MathWorld.
- Orthocenter of a triangle With interactive animation
- Animated demonstration of orthocenter construction Compass and straightedge.
- Fagnano's Problem by Jay Warendorff, Wolfram Demonstrations Project.