Orthocenter

Orthocenter (from other Greek: ὀρθός “straight line”) is the point of intersection of the heights of a triangle or their extensions. Traditionally denoted by the Latin letter $H$ ${\ displaystyle H}$ $H$ . Depending on the type of triangle, the orthocenter can be located inside the triangle (acute-angled), outside it (obtuse-angled) or coincide with a vertex (in a rectangular one - coincides with the vertex at right angles). The orthocenter refers to the remarkable points of the triangle and is listed in the encyclopedia of the centers of the triangle as point X (4).

Content

Properties

If the four points A, B, C, D point D is the intersection point of the heights of the triangle ABC , then any of the four points is the orthocenter of the triangle formed by the other three points. Such a four is sometimes called an orthocentric point system .
- Moreover, for any partition of the set of the orthocentric system of points { A, B, C, D } into two pairs, for example, { B, C } and { A, D } or for any other similar partition, the two segments of lines with ends at given points of the sets (in our case BC is perpendicular to AD ) regardless of the choice of these two pairs.
- The radii of the circles passing through any three points of the orthocentric system are equal (a consequence of Hamilton's theorem for the Euler circle ). They are often called Johnson circles.
- The last statement can be formulated as follows: Three segments of lines connecting the orthocenter with the vertices of an acute-angled triangle divide it into three triangles having equal radii of the circumscribed circles (a consequence of Hamilton's theorem for the Euler circle ). Moreover, the same radius of these three circles is equal to the radius of the circle described near the original acute-angled triangle.

Points symmetric to the orthocenter relative to the sides lie on the circumscribed circle.

The orthocenter lies on one straight line with the centroid , the center of the circumscribed circle and the center of the circle of nine points (see Euler's line ).
The orthocenter of an acute-angled triangle is the center of a circle inscribed in its ortho -triangle.
The center of the circle circumscribed around the triangle serves as the orthocenter of the triangle with vertices in the middle of the sides of the triangle. The last triangle is called an additional triangle with respect to the first triangle.
The last property can be formulated as follows: The center of the circle circumscribed around the triangle serves as the orthocenter of the additional triangle .
Points symmetric to the orthocenter of a triangle relative to its sides lie on the circumscribed circle (see figure) ^[1] .
Points symmetric to the orthocenter of the triangle with respect to the midpoints of the sides also lie on the circumscribed circle and coincide with points diametrically opposite the corresponding vertices.
If a $O$ {\ displaystyle O} Is the center of the circumscribed circle $△$ $A$ $B$ $C$ {\ displaystyle \ triangle ABC} then $O$ $H$ $\to$ $=$ $O$ $A$ $\to$ $+$ $O$ $B$ $\to$ $+$ $O$ $C$ $\to$ {\ displaystyle {\ overrightarrow {OH}} = {\ overrightarrow {OA}} + {\ overrightarrow {OB}} + {\ overrightarrow {OC}}} .
- $OH={\sqrt {R^{2}-8R^{2}\cos A\cos B\cos C}}={\sqrt {9R^{2}-(a^{2}+b^{2}+c^{2})}}$ ${\ displaystyle OH = {\ sqrt {R ^ {2} -8R ^ {2} \ cos A \ cos B \ cos C}} = {\ sqrt {9R ^ {2} - (a ^ {2} + b ^ {2} + c ^ {2})}}}$ ^[2] ^[3] where $R$ ${\ displaystyle R}$ - radius of the circumscribed circle ; $a, b, c$ ${\ displaystyle a, b, c}$ - the lengths of the sides of the triangle; $A, B, C$ ${\ displaystyle A, B, C}$ - internal angles of the triangle.
With isogonal conjugation, the orthocenter goes to the center of the circumscribed circle.
Any segment drawn from the orthocenter to the intersection with the circumscribed circle is always divided in two by the Euler circle . This follows from the fact that the orthocenter is the homothety center of these two circles with the coefficient $1/2$ ${\ displaystyle 1/2}$ .
Four pairwise intersecting lines, none of which pass through one point (four-sided), form four triangles at the intersection. Their orthocenters lie on one straight line ( on the Aubert line ).
If we assume that the triangle orthocenter divides the first height into parts of length: u and v , the second height into parts of length: w and x , the third height into parts of length: y and z , then uv = wx = yz ^[4] ^[5] .
The chain of equations in the last paragraph: uv = wx = yz , - essentially means that the three pairs of segments into which the orthocenter divides the three heights of an acute-angled triangle obey the rule of chords intersecting inside the circle, for example: uv = wx . It automatically follows from this that through the four ends of any two heights of an acute-angled triangle you can always draw a circle (the heights in it will be intersecting chords). It turns out that this statement remains valid for obtuse and rectangular triangles.
The distance from the side to the center of the circumscribed circle is equal to half the distance from the vertex opposite it to the orthocenter ^[6] ^[7] .
The sum of the squares of the distances from the vertices to the orthocenter plus the sum of the squares of the sides is equal to twelve squares of the radius of the circumscribed circle ^[8] .
Three bases of the heights of an acute-angled triangle or three projections of the orthocenter on the sides of the triangle form an ortho- triangle.

Orthocentric axis ( Orthic axis ) - trilinear polar of the orthocenter

The trilinear polar of the orthocenter is the orthocentric axis DEF (Orthic axis) (see. Fig.)

Four orthocenters of four triangles formed by four pairwise intersecting straight lines, none of which pass through one point, lie on one straight line ( Ober Square of a quadrilateral ) . Here, the same four triangles are used as in the construction of the Mikel point .

There is a Carnot formula ^[9] :

R+r=k_{a}+k_{b}+k_{c}={\frac {1}{2}}(d_{A}+d_{B}+d_{C}),

{\ displaystyle R + r = k_ {a} + k_ {b} + k_ {c} = {\ frac {1} {2}} (d_ {A} + d_ {B} + d_ {C}),}

Where $k_{a},k_{b},k_{c}$ ${\ displaystyle k_ {a}, k_ {b}, k_ {c}}$ - the distance from the center of the circumscribed circle, respectively, to the sides $a, b, c$ ${\ displaystyle a, b, c}$ triangle $d_{A},d_{B},d_{C}$ ${\ displaystyle d_ {A}, d_ {B}, d_ {C}}$ - the distance from the orthocenter, respectively, to the peaks $A, B, C$ ${\ displaystyle A, B, C}$ the triangle.

The distance from the center of the circumscribed circle to the side $a$ ${\ displaystyle a}$ equally:

k_{a}={\frac {a}{2\,\mathrm {tg} \,A}};

{\ displaystyle k_ {a} = {\ frac {a} {2 \, \ mathrm {tg} \, A}};}

distance from the orthocenter to the top $A$ ${\ displaystyle A}$ equally:

d_{A}={\frac {a}{\mathrm {tg} \,A}}.

{\ displaystyle d_ {A} = {\ frac {a} {\ mathrm {tg} \, A}}.}

Notes

↑ Honsberger, 1995 , p. 18.
↑ 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
↑ Smith, Geoff, and Leversha, Gerry, Euler and triangle geometry, Mathematical Gazette 91, November 2007, 436–452.
↑ Altshiller-Court, 2007 , p. 94.
↑ Honsberger, 1995 , p. 20.
↑ Altshiller-Court, 2007 , p. 99.
↑ Honsberger, 1995 , p. 17, 23.
↑ Altshiller-Court, 2007 , p. 102.
↑ Zetel S.I. New geometry of a triangle. A manual for teachers. 2nd edition. M .: Uchpedgiz, 1962. task on p. 120-125. paragraph 57, p. 73.

Literature

Ponarin I.P. Elementary geometry. In 2 vols. - M .: ICMMO , 2004 .-- S. 37-39. - ISBN 5-94057-170-0 .
Nathan Altshiller-Court. College geometry: an introduction to the modern geometry of the triangle and the circle . - Dover Publications, Inc., 2007. - ISBN 0-486-45805-9 .
Ross Honsberger. Episodes in Nineteenth and Twentieth Century Euclidean Geometry . - Mathematical Association of America , 1995. - Vol. 37. - P. 17-26. - (New Mathematical Library). - ISBN 0-88385-639-5 (Vol. 37). - ISBN 0-88385-600-X (complete set).

Links

Living drawing

[_81b62ededc90fe59-1] Honsberger, 1995 , p. 18.

[2] 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

[SL-3] Smith, Geoff, and Leversha, Gerry, Euler and triangle geometry, Mathematical Gazette 91, November 2007, 436–452.

[_ca89cffaa52cafa5-4] Altshiller-Court, 2007 , p. 94.

[_81b62bdedc90f978-5] Honsberger, 1995 , p. 20.

[_ca89cffaa52cafa8-6] Altshiller-Court, 2007 , p. 99.

[_416d825a92f1ce77-7] Honsberger, 1995 , p. 17, 23.

[_54c097e6aad6e8b9-8] Altshiller-Court, 2007 , p. 102.

[9] Zetel S.I. New geometry of a triangle. A manual for teachers. 2nd edition. M .: Uchpedgiz, 1962. task on p. 120-125. paragraph 57, p. 73.