Editing Altitude (triangle)

{{short description|Perpendicular line segment from a triangle's side to opposite vertex}}
[[Image:Projection formula (3).png|thumb|The altitude from A (dashed line segment) intersects the extended base at D (a point outside the triangle).]]

In [[geometry]], an '''altitude''' of a [[triangle]]  is a [[line segment]] through a given [[Vertex (geometry)|vertex]] (called ''[[apex (geometry)|apex]]'') and [[perpendicular]] to a [[line (geometry)|line]] containing the side or [[edge (geometry)|edge]] opposite the apex. This (finite) edge and (infinite) line extension are called, respectively, the ''[[base (geometry)|base]]'' and ''[[extended side|extended base]]'' of the altitude. The [[point (geometry)|point]] at 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" or "height", symbol {{mvar|h}}, is the distance between the foot and the apex.  The process of drawing the altitude from a vertex to the foot is known as ''dropping the altitude'' at that vertex. It is a special case of [[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 (symbol {{mvar|b}}) equals the triangle's area: {{mvar|A}}{{=}}{{mvar|h}}{{mvar|b}}/2. 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 [[isosceles triangle]] (a triangle with two [[congruence (geometry)|congruent]] sides), the altitude having the incongruent side as its base will have the [[midpoint]] of that side as its foot. Also the altitude having the incongruent side as its base will be the [[angle bisector]] of the vertex angle.

In a [[right triangle]], the altitude drawn to the [[hypotenuse]] {{mvar|c}} divides the hypotenuse into two segments of lengths {{mvar|p}} and {{mvar|q}}. If we denote the length of the altitude by {{mvar|h{{sub|c}}}}, we then have the relation
:<math>h_c=\sqrt{pq} </math>&nbsp; ([[Geometric mean theorem]]; see [[#Right_triangle|Special Cases]], [[inverse Pythagorean theorem]])

[[File:Rtriangle.svg|thumb|right|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.

==Theorems==

The geometric altitude figures prominently in many important theorems and their proofs. For example, besides those theorems listed below, the altitude plays a central role in proofs of both the [[Law_of_sines#Proof|Law of sines]] and [[Law_of_cosines#Using_the_Pythagorean_theorem|Law of cosines]].

===Orthocenter===
{{excerpt|Orthocenter}}

===Altitude in terms of the sides===

For any triangle with sides {{mvar|a, b, c}} and [[semiperimeter]] <math>s = \tfrac12(a+b+c),</math> the altitude from side {{mvar|a}} (the base) is given by

:<math>h_a=\frac{2\sqrt{s(s-a)(s-b)(s-c)}}{a}.</math>

This follows from combining [[Heron's formula]] for the area of a triangle in terms of the sides with the area formula <math>\tfrac{1}{2} \times \text{base} \times \text{height},</math> where the base is taken as side {{mvar|a}} and the height is the altitude from the vertex {{mvar|A}} (opposite side {{mvar|a}}).

By exchanging {{mvar|a}} with {{mvar|b}} or {{mvar|c}}, this equation can also used to find the altitudes {{mvar|h<sub>b</sub>}} and {{mvar|h<sub>c</sub>}}, respectively.

===Inradius theorems===

Consider an arbitrary triangle with sides {{mvar|a, b, c}} and with corresponding
altitudes {{mvar|h{{sub|a}}, h{{sub|b}}, h{{sub|c}}}}. The altitudes and the [[Incircle_and_excircles#Radius|incircle radius]] {{mvar|r}} are related by<ref>{{cite journal | last1 = Andrica | first1 = Dorin | last2 = Marinescu | first2 = Dan Ştefan | title = New Interpolation Inequalities to Euler's R ≥ 2r | journal = [[Forum Geometricorum]] | volume = 17 | year = 2017 | pages = 149–156 | url = http://forumgeom.fau.edu/FG2017volume17/FG201719.pdf | archive-url = https://web.archive.org/web/20180424054714/http://forumgeom.fau.edu/FG2017volume17/FG201719.pdf | archive-date = 2018-04-24 | url-status = dead}}</ref>{{rp|Lemma 1}}

:<math>\displaystyle \frac{1}{r}=\frac{1}{h_a}+\frac{1}{h_b}+\frac{1}{h_c}.</math>

===Circumradius theorem===

Denoting the altitude from one side of a triangle as {{mvar|h{{sub|a}}}}, the other two sides as {{mvar|b}} and {{mvar|c}}, and the triangle's [[circumradius]] (radius of the triangle's circumscribed circle) as {{mvar|R}}, the altitude is given by<ref>{{harvnb|Johnson|2007|loc=p. 71, Section 101a}}</ref>

:<math>h_a=\frac{bc}{2R}.</math>

===Interior point===

If {{math|''p''{{sub|1}}, ''p''{{sub|2}}, ''p''{{sub|3}}}} are the perpendicular distances from any point {{mvar|P}} to the sides, and {{math|''h''{{sub|1}}, ''h''{{sub|2}}, ''h''{{sub|3}}}} are the altitudes to the respective sides, then<ref>{{harvnb|Johnson|2007|loc=p. 74, Section 103c}}</ref>

:<math>\frac{p_1}{h_1} +\frac{p_2}{h_2} + \frac{p_3}{h_3} = 1.</math>

===Area theorem===

Denoting the altitudes of any triangle from sides {{mvar|a, b, c}} respectively as {{mvar|h{{sub|a}}, h{{sub|b}}, h{{sub|c}}}}, and denoting the semi-sum of the reciprocals of the altitudes as <math>H = \tfrac{h_a^{-1} + h_b^{-1} + h_c^{-1}}{2}</math> we have<ref>Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle," ''Mathematical Gazette'' 89, November 2005, 494.</ref>

:<math>\mathrm{Area}^{-1} = 4 \sqrt{H(H-h_a^{-1})(H-h_b^{-1})(H-h_c^{-1})}.</math>

===General point on an altitude===

If {{mvar|E}} is any point on an altitude {{mvar|{{overline|AD}}}} of any triangle {{math|△''ABC''}}, then<ref name=Posamentier>[[Alfred S. Posamentier]] and Charles T. Salkind, ''Challenging Problems in Geometry'', Dover Publishing Co., second revised edition, 1996.</ref>{{rp|77–78}}
:<math>\overline{AC}^2 + \overline{EB}^2 = \overline{AB}^2 + \overline{CE}^2.</math>

===Triangle inequality===

Since the area of the triangle is <math>\tfrac12 a h_a = \tfrac12 b h_b = \tfrac12 c h_c</math>, the triangle inequality <math>a < b+ c</math> implies<ref>Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle", ''Mathematical Gazette'' 89 (November 2005), 494.</ref> 
:<math>\frac1{h_a} < \frac1{h_b}+ \frac1{h_c}</math>.

===Special cases===

====Equilateral triangle====

From any point {{mvar|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====
{{right_angle_altitude.svg}}
[[File:inverse_pythagorean_theorem.svg|thumb|Comparison of the inverse Pythagorean theorem with the Pythagorean theorem]]
In a right triangle with legs {{mvar|a}} and {{mvar|b}} and hypotenuse {{mvar|c}}, each of the legs is also an altitude: {{tmath|1= h_a = b}} and {{tmath|1= h_b = a}}. The third altitude can be found by the relation<ref>Voles, Roger, "Integer solutions of <math>a^{-2}+b^{-2}=d^{-2}</math>," ''Mathematical Gazette'' 83, July 1999, 269–271.</ref><ref>Richinick, Jennifer, "The upside-down Pythagorean Theorem," ''Mathematical Gazette'' 92, July 2008, 313–317.</ref>

:<math>\frac{1}{h_c ^2} = \frac{1}{h_a ^2}+\frac{1}{h_b ^2} = \frac{1}{a^2}+\frac{1}{b^2}.</math>

This is also known as the [[inverse Pythagorean theorem]].

Note in particular:
:<math>\begin{align}
\tfrac{1}{2} AC \cdot BC &= \tfrac{1}{2} AB \cdot CD \\[4pt]
         CD &= \tfrac{AC \cdot BC}{AB} \\[4pt]
\end{align}</math>

==See also==
*[[Median (geometry)]]

{{clear}}

==Notes==
{{Reflist|30em}}

==References==
* {{citation |last=Altshiller-Court |first=Nathan |author-link=Nathan Altshiller Court |year=2007 |orig-year=1952 |title=College Geometry |publisher=Dover }}
* {{citation |last1=Berele |first1=Allan |last2=Goldman |first2=Jerry |year=2001 |title=Geometry: Theorems and Constructions |publisher=Prentice Hall |isbn=0-13-087121-4 }}
* {{cite web | last=Bogomolny |first=Alexander |author-link= Alexander Bogomolny |website=Cut the Knot |title=Existence of the Orthocenter |url=http://www.cut-the-knot.org/triangle/altitudes.shtml |access-date=2022-12-17 }}
* {{citation |last=Johnson |first=Roger A. |year=2007 |orig-year=1960 |title=Advanced Euclidean Geometry |publisher=Dover |isbn=978-0-486-46237-0 }}
* {{citation |last=Smart |first=James R. |year=1998 |title=Modern Geometries |edition=5th |publisher=Brooks/Cole |isbn=0-534-35188-3}}

==External links==
* {{MathWorld|title=Altitude|urlname=Altitude}}

[[Category:Straight lines defined for a triangle]]

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