1. Orthocenter

Three altitudes intersecting at the orthocenter. https://handwiki.org/wiki/index.php?curid=2069295

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 (i.e. does not have an angle greater than or equal to a right angle). 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 a = |BC|, b = |CA|, c = |AB| be the side lengths. The orthocenter has trilinear coordinates^[3]

[math]\displaystyle{ \sec A:\sec B:\sec C = \cos A-\sin B \sin C:\cos B-\sin C \sin A:\cos C-\sin A\sin B, }[/math]

and barycentric coordinates

[math]\displaystyle{ \displaystyle (a^2+b^2-c^2)(a^2-b^2+c^2) : (a^2+b^2-c^2)(-a^2+b^2+c^2) : (a^2-b^2+c^2)(-a^2+b^2+c^2) }[/math]

[math]\displaystyle{ =\tan A:\tan B:\tan C. }[/math]

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 acute triangle's interior, on the right-angled vertex of a right triangle, and exterior to an obtuse triangle.

In the complex plane, let the points A, B and C represent the numbers [math]\displaystyle{ z_A }[/math], [math]\displaystyle{ z_B }[/math] and, respectively, [math]\displaystyle{ z_C }[/math] and assume that the circumcenter of triangle ABC is located at the origin of the plane. Then, the complex number

[math]\displaystyle{ z_H=z_A+z_B+z_C }[/math]

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

[math]\displaystyle{ \vec{OH}=\sum\limits_{\scriptstyle\rm cyclic}\vec{OA},\qquad2\cdot\vec{HO}=\sum\limits_{\scriptstyle\rm cyclic}\vec{HA}. }[/math]

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

1.1. Properties

Let D, E, and F denote the feet of the altitudes from A, B, and 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]

[math]\displaystyle{ AH \cdot HD = BH \cdot HE = CH \cdot HF. }[/math]

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.)

[math]\displaystyle{ \frac{HD}{AD} + \frac{HE}{BE} + \frac{HF}{CF} = 1. }[/math]

• 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]

[math]\displaystyle{ \frac{AH}{AD} + \frac{BH}{BE} + \frac{CH}{CF} = 2. }[/math]

• The isogonal conjugate of the orthocenter is 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.

1.2. Relation with Circles and Conics

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

[math]\displaystyle{ a^2+b^2+c^2+AH^2+BH^2+CH^2 = 12R^2. }[/math]

In addition, denoting r as the radius of the triangle's incircle, r_a, r_b, and r_c as the radii of its excircles, and R again as the radius of its circumcircle, the following relations hold regarding the distances of the orthocenter from the vertices:^[14]

[math]\displaystyle{ r_a+r_b+r_c+r=AH+BH+CH+2R, }[/math]

[math]\displaystyle{ r_a^2+r_b^2+r_c^2+r^2=AH^2+BH^2+CH^2+(2R)^2. }[/math]

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

[math]\displaystyle{ HD = DP. }[/math]

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 circumconic passing through the orthocenter of a triangle is a rectangular hyperbola.^[16]

1.3. Relation to Other Centers, the Nine-Point Circle

The orthocenter H, the centroid G, 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]

[math]\displaystyle{ OH=2NH, }[/math]

[math]\displaystyle{ 2OG=GH. }[/math]

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:

[math]\displaystyle{ HI \lt HG, }[/math]

[math]\displaystyle{ HG \gt IG. }[/math]

In terms of the sides a, b, c, inradius r and circumradius R,^[19]

[math]\displaystyle{ OH^2 = R^2 -8R^2 \cos A \cos B \cos C=9R^2-(a^2+b^2+c^2), }[/math]^{[20]:p. 449}

[math]\displaystyle{ HI^2 = 2r^2 -4R^2 \cos A \cos B \cos C. }[/math]

2. Orthic Triangle

Triangle abc (respectively, DEF in the text) is the orthic triangle of triangle ABC. https://handwiki.org/wiki/index.php?curid=2058426

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

• D = 0 : sec B : sec C
• E = sec A : 0 : sec C
• F = sec A : sec B : 0.

The extended sides of the orthic triangle meet the opposite extended sides of its reference triangle at three collinear points.^[21][22][23]

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 L_A be the line tangent to the circumcircle of triangle ABC at vertex A, and define L_B and L_C analogously. Let A" = L_B ∩ L_C, B" = L_C ∩ L_A, C" = L_C ∩ L_A. The tangential triangle is A"B"C", whose sides are the tangents to triangle ABC's circumcircle at its vertices; it is homothetic to the orthic triangle. The circumcenter of the tangential triangle, and the 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

• A" = −a : b : c
• B" = a : −b : c
• C" = a : b : −c.

For more information on the orthic triangle, see here.

3. Some Additional Altitude Theorems

3.1. Altitude in Terms of the Sides

For any triangle with sides a, b, c and semiperimeter s = (a + b + c) / 2, the altitude from side a is given by

[math]\displaystyle{ 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 (1/2)×base×height, where the base is taken as side a and the height is the altitude from A.

3.2. Inradius Theorems

Consider an arbitrary triangle with sides a, b, c and with corresponding altitudes h_a, h_b, and h_c. The altitudes and the incircle radius r are related by^{[29]:Lemma 1}

[math]\displaystyle{ \displaystyle \frac{1}{r}=\frac{1}{h_a}+\frac{1}{h_b}+\frac{1}{h_c}. }[/math]

3.3. Circumradius Theorem

Denoting the altitude from one side of a triangle as h_a, 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]

[math]\displaystyle{ h_a=\frac{bc}{2R}. }[/math]

3.4. Interior Point

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

[math]\displaystyle{ \frac{p_1}{h_1} +\frac{p_2}{h_2} + \frac{p_3}{h_3} = 1. }[/math]

3.5. Area Theorem

Denoting the altitudes of any triangle from sides a, b, and c respectively as [math]\displaystyle{ h_a }[/math], [math]\displaystyle{ h_b }[/math], and [math]\displaystyle{ h_c }[/math], and denoting the semi-sum of the reciprocals of the altitudes as [math]\displaystyle{ H = (h_a^{-1} + h_b^{-1} + h_c^{-1})/2 }[/math] we have^[32]

[math]\displaystyle{ \mathrm{Area}^{-1} = 4 \sqrt{H(H-h_a^{-1})(H-h_b^{-1})(H-h_c^{-1})}. }[/math]

3.6. General Point on an Altitude

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

[math]\displaystyle{ AC^2+EB^2=AB^2+CE^2. }[/math]

3.7. Special Case Triangles

Equilateral triangle

For 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

 

https://handwiki.org/wiki/index.php?curid=2036389. The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments the hypotenuse is split into. Using Pythagoras' theorem on the 3 triangles of sides (p + q, r, s), (r, h, p) and (s, h, q),
[math]\displaystyle{ \begin{align} (p + q)^2 \quad &= \quad\;\, r^2 \quad + \quad s^2 \\ p^2 \!+ 2pq + q^2 &= \left(h^2 \!\!+\! p^2\right) \!+\! \left(h^2 \!\!+\! q^2\right) \\ 2pq \qquad\, &= 2h^2 \quad \therefore h = \sqrt{pq} \\ \end{align} }[/math]

In a right triangle the three altitudes h_a, h_b, and h_c (the first two of which equal the leg lengths b and a respectively) are related according to^[34][35]

[math]\displaystyle{ \frac{1}{h_a ^2}+\frac{1}{h_b ^2}=\frac{1}{h_c ^2}. }[/math]

4. History

The theorem that the three altitudes of a triangle meet in a single point, the orthocenter, was first proved in a 1749 publication by William Chapple.^[36]

The content is sourced from: https://handwiki.org/wiki/Altitude_(triangle)