**Remarkable points of a triangle** are points whose location is uniquely determined by a triangle and does not depend on the order in which the sides and vertices of the triangle are taken.

Usually they are located inside the triangle, but this is not necessary. In particular, the point of intersection of heights may be outside the triangle. For other remarkable points of the triangle, see the Encyclopedia of Triangle Centers .

## Examples

Remarkable points of the triangle are

- Intersection points:
- Median - centroid , center of gravity (mass );
- Bisectors - the center or the center of the inscribed circle ;
- Anti - bisector - the center of anti-bisector;
- The outer angular bisectors are the center of the non-inscribed circle ;
- Heights - orthocenter ;
- Middle perpendiculars - the center of the circumscribed circle ;
- Symedian - Lemoine Point ;
- The bisector of the middle triangle (its center) is the Speaker Center ;
- The triangular jib is also the Speaker Center;
- Three (or even two) circles, built as on diameter, on the segment connecting the bases of the internal and external bisectrix ejected from the same angle are two Apollonian points ;
- The segments connecting the vertices of the triangle:
- with the points of tangency of opposite sides and the inscribed circle , the Gergonne point ;
- with points of tangency of opposite sides and extra-written circles , the Nagel point ;
- with the corresponding free vertices of equilateral triangles built on the sides of the triangle (outside) - the first Torricelli point ;
- with the corresponding free vertices of regular triangles constructed inside the triangle - the second Torricelli point ;
- c corresponding free vertices of triangles similar to the original triangle and built on its sides are Brocard points ;

## Minimax points of a triangle

**The minimax (extremal) points of a triangle** are the points at which the minimum of a certain function is reached, for example, the sum of the degrees of the distances to the sides or vertices of the triangle ^{[1]} .

The minimax points of a triangle are:

- The point of intersection of the three medians , which has the smallest sum of squares of distances to the vertices of the triangle ( Leibniz theorem ).
- The intersection point of three medians of a triangle is the only point of a triangle such that the three Chevians drawn through it divide the sides of the triangle into six segments by their ends. Moreover, the product of the lengths of three of these six segments that have no common ends is
*maximal*^{[2]} - Torricelli Point (first), which has the smallest sum of distances to the vertices of a triangle with angles of not more than 120 degrees.
- Lemoine point , which has the smallest sum of squares of distances to the sides of a triangle.
- The bases of the heights of an acute triangle form an ortho -triangle having the smallest perimeter of all triangles inscribed in a given triangle.

## Iso-points and iso-lines of a triangle

**The iso-points** are the points of the triangle, giving any equal parameters of the three triangles, which are formed when the iso-point is joined by segments with three vertices of the triangle ^{[3]} . As a result, a figure of the “ dragon eye ” type is formed (see fig.)

### From the point of the triangle, forming a figure like a " dragon eye "

From-points of a triangle of this type are:

- orthocenter (gives three triangles with equal radii of three circles described around them),
- intersection point of medians (gives three triangles with equal areas)
- incentre (gives three triangles with equal heights)
- the center of the circumscribed circle (gives three isosceles triangles with equal pairs of sides),
**point of equal perimeters***P*or*isoperimetric point*(gives three triangles with equal perimeters, see http://faculty.evansville.edu/ck6/tcenters/recent/isoper.html ),- Torricelli point (first) (gives three triangles with equal obtuse angles of 120 degrees).

### From-points of a triangle forming a trefoil (knot) figure

From-points of a triangle of this type are (see fig.):

- Center of Shpiker
*S*is the intersection point of straight lines*AX*,*BY*and*CZ*, where*XBC*,*YCA*and*ZAB are*similar, isosceles and equally spaced, built on the sides of the triangle*ABC*outside, having the same angle at the base$\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}[\mathrm{t}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}(A/2)\mathrm{t}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}(B/2)\mathrm{t}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}(C/2)]$^{[4]}. - The first point of Napoleon$N}_{\mathrm{one}$ , like the
*Spiker center*, is the intersection point of the lines*AX*,*BY*and*CZ*, where*XBC*,*YCA*and*ZAB are*similar, isosceles and equally spaced, built on the sides of the triangle*ABC*outside, having the same angle at the base of 30 degrees.

### From-points of a triangle forming a tradescantia flower type

From the point of a triangle, forming a figure of the “Flower of Tradescantia” type (see fig.) The following:

- the intersection point of the medians forms three small quadrilaterals with equal squares in three small segments of the Chevian.
- the intersection point of the bisectors is formed by three perpendiculars to the three sides of the triangle, three quadrilaterals — the deltoid with two identical adjacent sides. Another pair of equal adjacent sides is generally different for everyone. All three deltoids have a pair of equal opposite angles of 90 degrees. They are the inscribed quadrangles.
- Three circles drawn inside the triangle through Mikel point intersect the sides of the triangle at three points. Three chords drawn through the Mikel point and three intersection points of three circles with three different sides of the triangle form equal angles with the sides.

### Other triangles from point to point

From-points of a triangle of this type are:

- Lemoine point (a point of equal antiparallel) - a point with a property: 3 antiparallels drawn through it (lines antiparallel to 3 sides of a triangle) give 3 segments of equal length inside a triangle.
- Equal Parallelians Point
^{[5]}. In a sense, is similar to the Lemoine point . The point has the property: 3 parallels drawn through it (lines parallel to 3 sides of a triangle) give 3 segments of equal length inside a triangle. **Skutina points**are points equal to the Chevian of a triangle.*Theorem Skutina*states that three segments of lines or Chevians drawn inside a triangle through its three vertices and through any focus of the described*Steiner ellipse are*equal to each other. These tricks are often called*Skutin points*.

### Izo direct

**The i-lines** ( *i-lines* ) of a triangle are straight lines that cut the given triangle into two triangles having any equal parameters ^{[3]} . From the straight lines of the triangle are:

- The median of the triangle divides the opposite side in half and cuts the triangle into two triangles with equal squares.
- The bisector of the triangle halves the angle from the top of which it comes out.
- The height of the triangle intersects the opposite side (or its extension) at a right angle (that is, it forms two equal angles with a side on either side of it) and cuts the triangle into two triangles with equal (right) angles.
- Symedian is a locus of points inside a triangle, emerging from one vertex, giving two equal segments, antiparallel to two sides intersecting at this vertex, and bounded by three sides.
- Triangle jib splits the
*perimeter in half*. A triangle jib is a segment, one end of which is in the middle of one of the sides of the triangle, the other end is on one of the two remaining sides. In addition, the jib is parallel to one of the bisectors of the angle. Each of the jibs passes through the center of mass of the perimeter of the triangle ABC, so that all three jibs intersect at the center of the spikeer . - It also splits the
*perimeter in half, the*segment connecting the point of tangency of the side of the triangle and the extra-circumscribed circle with the vertex opposite to this side. Three such segments of a triangle drawn from its three vertices intersect at the Nagel point . In other words, this segment is the*Chevian of*the Nagel point . (*Chevian Nagel points*in English literature are sometimes called a*splitter*or*splitter in half the perimeter*. They also refer to the*splitter*to the*splitter*). *Equalizer*(Equalizer) or*Equalizer*(*Equalizer*) - a straight line segment that cuts a triangle into two figures of equal areas and perimeters simultaneously^{[6]}- A little bit about the
*equalizer*(equalizer). Any straight line (*equalizer*) passing through the triangle and dividing the area of the triangle and the perimeter in half, passes through the center of the inscribed circle. There may be three such lines, two or one.^{[7]}

### Note on iso-straight triangles

The English literature introduces the concept of *bisection* (Bisection), as the division of something into two equal parts. For example, an isosceles triangle into two equal, straight line into two equal, flat angles into two equal. The corresponding lines will be a special case of iso-straight (is-lines) triangle.

### Straight n

An important particular case of iso-lines is the so-called *straight lines n of a* triangle. *The line n of the* triangle, starting from its vertex, divides the opposite side with respect to the *nth* degrees of the two sides adjacent to it ^{[8]} . Important special cases of *direct n* are:

- median (
*n*= 0) - bisector (bisector) (
*n*= 1) - anti-bisector (
*n*= -1) - symmedian (
*n*= 2) - straight cubes (
*n*= 3)

For *straight lines n of a* triangle it is very easy to find some properties in general form. For example, for a *line n, a* *straight line (2-n)* will be isogonally conjugate, and a *straight minus n* will be isotomically conjugate.

## Note

The barycentric coordinates of the center, written through the sides (or trigonometric functions of the angles) of the triangle, make it possible to translate many problems of the centers of the triangle into an algebraic language. For example, find out if two definitions define the same center or if the three center data lie on one straight line.

You can use the trilinear coordinates of the center, very simply associated with the barycentric coordinates . However, for example, isogonal conjugate points in trilinear coordinates are expressed more simply.

## Variations and generalizations

- Consider a pair of centers. for example
- Brocard points .
- Apollonia points . For any non-degenerate ABC triangle, one can construct the Apollonian circle to the side AB, passing through point C. The circles constructed in this way to three sides will intersect at two points - the inner and outer Apollonius, respectively.

## Recently opened points (centers) of a triangle

- Point of congruence Iffa ( eng. Yff Center of Congruence )
^{[9]} - Perspector Gossard ( Eng. Gossard Perspector )
^{[10]} - Midpoint ( eng. Mittenpunkt )
^{[11]} - 1st and 2nd points of Ajima-Malfatti ( English 1ST AND 2ND Ajima-Malfatti Points )
^{[12]} - Apollonia Point - not to be confused with Apollonia points
^{[13]} - Bailey Point ( eng. Bailey Point )
^{[14]} - Gofstadter points ( English Hofstadter Points )
^{[15]} - Iso-skeleton-congruent point ( eng. Congruent Isoscelizers Point )
^{[16]} - The 1st and 2nd Morley points associated with the Morley triangle ( 1ST AND 2ND Morley Centers )
^{[17]} - Parry Point ( English Parry Point )
^{[18]} - Point of equal perimeter and equal detour ( Isoperimetric Point and Equal Detour Point )
^{[19]} - Equal Parallel Points ( Eng. Equal Parallelians Point )
^{[20]} - Schiffler Point ( Eng. Schiffler Point )
^{[21]} - Exeter Point
^{[22]}

## Notes

- ↑
*Starikov, V.N.*Studies in Geometry // Collection of publications of the scientific journal*Globus*based on materials from the Vth International Scientific and Practical Conference "Achievements and Problems of Modern Science", St. Petersburg: a collection with articles (standard level, academic level) . S.-P .: Scientific journal*Globus*, 2016. P. 97. - ↑
*Zetel, S. I.*New triangle geometry. Manual for teachers. 2nd edition. M.: Uchpedgiz, 1962. task on p. 12. - ↑
^{1}^{2}Starikov V.N. Notes on geometry // Scientific search: humanitarian and socio-economic sciences: collection of scientific papers. Issue 1 / Ch. Ed. Romanova I .V Cheboksary: CIND “INet”, 2014. p. 37, left column, last paragraph - ↑
*Weisstein, Eric W.*Kiepert Hyperbola (Eng.) On Wolfram MathWorld . - ↑ Equal Parallelians Point
- ↑ Kodokostas, Dimitrios (2010), " Triangle equalizers ",
*Mathematics Magazine*T. 83 (2): 141–146 , DOI 10.4169 / 002557010X482916 . - ↑
*Dimitrios Kodokostas.*Triangle Equalizers // Mathematics Magazine. - 2010. - Vol. 83, April . - p. 141-146. . - ↑
*Zetel, S. I.*New triangle geometry. Manual for teachers. 2nd edition. M.: Uchpedgiz, 1962. task on p. 120-125. paragraphs 109-113. - ↑ Yff Center Of Congruence
- ↑ Gossard Perspector
- ↑ Mittenpunkt
- ST 1ST AND 2ND AJIMA-MALFATTI POINTS
- ↑ Apollonius Point
- ↑ Bailey Point
- ↑ Hofstadter Points
- ↑ Congruent Isoscelizers Point
- ↑ Morley Centers
- ↑ Parry Point
- ↑ Isoperimetric Point And Equal Detour Point
- ↑ Equal Parallelians Point
- ↑ Schiffler Point
- ↑ Exeter Point

## Literature

- Encyclopedia for children. T. 11. Mathematics / Chapter. ed. M.D. Aksenov. - M .: Avanta +, 2001. - 688 pp., Ill.
*A. G. Myakishev.*Elements of the geometry of a triangle . - M .: MTSNMO, 2002.