Usually they are located inside the triangle, but this is not necessarily the case. In particular, the point of intersection of the altitudes may lie outside the triangle.
AN, BM — bisectors, 𝑂 — their point of intersection.
Is CK a bisector? If point 𝑂 is equidistant from sides AB, AC and from sides BA, BC, then it lies on the bisector of the angle
This point is the center of the circle inscribed in the triangle, and it is always located inside the triangle.
The second remarkable point of a triangle — the point of intersection of the perpendicular bisectors of the sides of the triangle
Theorem 6. The perpendicular bisectors of the sides of a triangle intersect at one point.
Suppose point O is the point of intersection of two perpendicular bisectors of sides AB, BC. It is equidistant from points A and B, and from points B and C. Consequently, it lies on the perpendicular bisector of side AC, since it is equidistant from its endpoints.
This point is the center of the circle circumscribed about the triangle; it lies inside triangles with acute angles, outside a triangle with an obtuse angle, and on the hypotenuse of a right triangle.
The third remarkable point of a triangle — the point of intersection of the medians
Theorem 7. The medians of a triangle intersect at one point, which divides each median in a ratio of 2:1, counting from the vertex.
The point of intersection of the medians is the centroid of the triangle.
The fourth remarkable point of a triangle — the point of intersection of the altitudes of the triangle
Theorem 8. The altitudes of a triangle, or their extensions, intersect at one point.
The point of intersection of the altitudes is called the orthocenter of the triangle.
In 1765, the German mathematician Euler proved that in any triangle the orthocenter, the centroid, and the center of the circumscribed circle lie on one line, later named the Euler line.
In the 1820s, French mathematicians Poncelet, Brianchon, and others independently established the following theorem: the feet of the medians, the feet of the altitudes, and the midpoints of the altitude segments connecting the orthocenter to the vertices of the triangle lie on one and the same circle.
Examples
Centroid — the point of intersection of the medians
Orthocenter — the point of intersection of the altitudes
The remarkable points of a triangle are
- Points of intersection:
- medians — the centroid, the center of gravity (mass);
- bisectors — the incenter, or the center of the inscribed circle;
- antibisectors — the center of the antibisectors;
- bisectors of exterior angles — the center of an excircle;
- altitudes — the orthocenter;
- perpendicular bisectors — the center of the circumscribed circle;
- symmedians — the Lemoine point;
- bisectors of the medial triangle (its incenter) — the Spieker center;
- cleavers of the triangle — also the Spieker center;
- three (or even two) circles constructed, as on a diameter, on the segment joining the feet of the internal and external bisectors drawn from one angle, — the two Apollonius points;
- segments connecting the vertices of the triangle:
- with the points of tangency of the opposite sides and the inscribed circle — the Gergonne point;
- with the points of tangency of the opposite sides and the excircles — the Nagel point;
- with the corresponding free vertices of equilateral triangles constructed on the sides of the triangle (outward) — the first Torricelli point;
- with the corresponding free vertices of regular triangles constructed inward on the triangle — the second Torricelli point;
- with the corresponding free vertices of triangles similar to the original triangle and constructed on its sides — the Brocard points;
Minimax points of a triangle
Minimax (extremal) points of a triangle are points at which the minimum of some function is attained, for example, the sum of powers of the distances to the sides or vertices of the triangle .
The minimax points of a triangle are:
- The point of intersection of the three medians, which has the smallest sum of squared distances to the vertices of the triangle (Leibniz's theorem).
- The point of intersection of the three medians of a triangle is the unique point of the triangle such that the three cevians drawn through it divide the sides of the triangle at their endpoints into six segments. Moreover, the product of the lengths of three of these six segments that have no common endpoints is maximal
- The Torricelli point (the first one), which has the smallest sum of distances to the vertices of a triangle with angles no greater than
.
- the Lemoine point, which has the smallest sum of squared distances to the sides of the triangle.
- The feet of the altitudes of an acute triangle form the orthic triangle, which has the smallest perimeter of all triangles inscribed in the given triangle.
Iso-points and iso-lines of a triangle
Iso-points are points of a triangle that give some equal parameters of the three triangles formed when the iso-point is joined by segments to the three vertices of the triangle . This results in a figure of the «dragon's eye» type (see figure)
Iso-points of a triangle forming a figure of the «dragon's eye» type
The iso-points of a triangle of this type are:
- the orthocenter (gives three triangles with three equal radii of the three circles circumscribed about them),
- the point of intersection of the medians (gives three triangles with three equal areas)
- the incenter (gives three triangles with three equal altitudes)
- the center of the circumscribed circle (gives three isosceles triangles with three equal pairs of sides),
- the point of equal perimeters
or isoperimetric point (gives three triangles with three equal perimeters ),
- the Torricelli point (the first one) (gives three triangles with three equal obtuse angles of ).
- The point that divides the triangle into three triangles with three equal radii of the inscribed circles
- The Spieker center of a triangle is the radical center of its three excircles (it has three pairs of equal tangents simultaneously to the three excircles).
Iso-points of a triangle forming a figure of the «Trefoil (knot)» type
The iso-points of a triangle of this type are (see figure):
- The Spieker center
is the point of intersection of the lines 
, 
and 
, where 
, 
and 
similar, isosceles, and similarly placed, constructed on the sides of triangle 
outward, having the same base angle 
.
- The first Napoleon point
, like the Spieker center, is the point of intersection of the lines , and , where , and similar, isosceles, and similarly placed, constructed on the sides of triangle outward, having the same base angle 
.
- Here one should list all the points lying on the Kiepert hyperbola.
Iso-points of a triangle forming a figure of the «Tradescantia flower» type
Stylized tradescantia flower
The iso-points of a triangle forming a figure of the «Tradescantia flower» type (see figure) are the following:
- the point of intersection of the medians forms, with three small cevian segments, three quadrilaterals with equal areas.
- the point of intersection of the bisectors forms, with three perpendiculars to the three sides of the triangle, three kite (deltoid) quadrilaterals with two adjacent sides equal that are the same for all three. The other pair of equal adjacent sides is generally different for each one. All three kites have a pair of equal opposite angles of {\displaystyle 90^{\circ }}
. They are bicentric quadrilaterals.
- Three circles drawn inside the triangle through the Miquel point intersect the sides of the triangle at three points. Three chords drawn through the Miquel point and the three points of intersection of the three circles with the three different sides of the triangle form equal angles with the sides.
Iso-points of a triangle forming a figure of the «Model of the surface of a curvilinear triangle» type (see figure)
Figure of the type: model of the surface of a curvilinear triangle
Stylized sign of the type "Model of the surface of a curvilinear triangle"
These points include:
- Points of the Euler circle
- Points in Thomsen's theorem
- Points in Tucker's theorem. If, in the figure for Thomsen's theorem, a similar 6-link broken line is drawn on the right below, successively alternating segments parallel, antiparallel, parallel, again antiparallel, again parallel to the current opposite side, and so on, then the last, 6th segment will return to the starting point, just as in Thomsen's theorem, and the broken line will close up. Tucker's theorem states that in this case the 6 points of the broken line lying on the sides of the triangle will lie on the Tucker circle
Iso-points of a triangle forming a sign of the type "Danger. Radioactive substances or ionizing radiation" (see fig.)
Sign "Danger. Radioactive substances or ionizing radiation"
The iso-points of a triangle of this type are:
- the Lemoine point (point of equal antiparallels) — a point with the property that the three antiparallels drawn through it (lines antiparallel to the three sides of the triangle) give three segments of equal length inside the triangle.
- the equal parallelians point (Equal Parallelians Point) . In a certain sense it is analogous to the Lemoine point. The point has the property that the three parallels drawn through it (lines parallel to the three sides of the triangle) give three segments of equal length inside the triangle.
- The Yff center of congruence (Yff Center of Congruence) [10]
- the point of intersection of the 3 antibisectors of the triangle. If 3 lines parallel to the sides of the triangle are drawn through this point, they cut off 3 equal inner (middle) segments on the sides of the triangle.
- Another formulation of the last statement: The segments of the sides of the triangle enclosed between the lines drawn through the antibisector center parallel to the three sides are equal to one another.
Other iso-points of a triangle forming cevians of general form
- Skutin points — points of equal cevians of a triangle. Skutin's theorem states that three line segments, or cevians, drawn inside a triangle through its three vertices and through any focus of the circumscribed Steiner ellipse, are equal to one another. These foci are often called Skutin points.
Iso-lines
Iso-lines (isolines) of a triangle are straight lines that cut a given triangle into two triangles having some equal parameters . The iso-lines of a triangle are:
- The median of a triangle bisects the opposite side and cuts the triangle into two triangles of equal area.
- The bisector (Bisector) of a triangle bisects the angle from whose vertex it emerges.
- The altitude of a triangle intersects the opposite side (or its extension) at a right angle (that is, it forms two equal angles with the side on both sides of it) and cuts the triangle into two triangles with equal (right) angles.
- The symmedian — the locus of points inside a triangle, emerging from one vertex, giving two equal segments antiparallel to the two sides meeting at that vertex and bounded by the three sides.
- The cleaver of a triangle bisects the perimeter. The cleaver of a triangle is a segment, one end of which lies at the midpoint of one of the sides of the triangle, and the other end lies on one of the two remaining sides. In addition, the cleaver is parallel to one of the angle bisectors. Each of the cleavers passes through the centroid of the perimeter of triangle ABC, so that all three cleavers meet at the Spieker center.
- The segment connecting the point where a side of the triangle touches the excircle to the vertex opposite that side also bisects the perimeter. Three such segments of the triangle, drawn from its three vertices, meet at the Nagel point. In other words, this segment is the cevian of the Nagel point. (The cevian of the Nagel point is sometimes called, in the English-language literature, a splitter (splitter) or a perimeter bisector. The cleaver is also classed among splitters.)
- The equalizer (equalizer), or leveler (balancer) — a straight-line segment that cuts a triangle into two figures of simultaneously equal areas and perimeters[11]
- A little about the equalizer (equalizer). Any line (equalizer) passing through a triangle and bisecting both the area and the perimeter of the triangle passes through the incenter. There may exist three, two, or one such lines.[12]
Remark on the iso-lines of a triangle
In the English-language literature the concept of bisection (Bisection) is introduced, meaning the division of something into two equal parts. For example, an isosceles triangle into two equal ones, a line segment into two equal ones, a plane angle into two equal ones. The corresponding lines will be a special case of the iso-lines (isolines) of a triangle.
X-lines 
An important special case of iso-lines are the so-called X-lines of a triangle. The X-line of a triangle, emerging from one of its vertices, divides the opposite side in the ratio of the -th powers of the two adjacent sides[13]. Important special cases of the -lines are:
- the median (
),
- the bisector (bisector) (
),
- the antibisector (
),
- the symmedian (
),
- the cube line (
).
For the -lines of a triangle it is very easy to find some properties in general form. For example, for the line the isogonal conjugate is the line 
, and the isotomic conjugate is the line 
.
Remark
Barycentric coordinates of a center, written in terms of the sides (or trigonometric functions of the angles) of the triangle, make it possible to translate many problems about triangle centers into algebraic language. For example, to determine whether two definitions specify the same center, or whether three given centers lie on one line.
One can also use trilinear coordinates of a center, which are very simply related to barycentric coordinates. However, for example, isogonally conjugate points are expressed more simply in trilinear coordinates.
Variations and generalizations
- Pairs of centers are considered. For example,
- the Brocard points;
- The Apollonius points. For any non-degenerate triangle one can construct the Apollonius circle for side
, passing through the point 
. The circles constructed in this way for the three sides will intersect at two points — the inner and outer Apollonius points, respectively.
Recently discovered points (centers) of a triangle
Main source: http://faculty.evansville.edu/ck6/tcenters/index.html
- The Yff center of congruence (Yff Center of Congruence)
- The Gossard perspector (Gossard Perspector)[
- The Mittenpunkt (Mittenpunkt)
- The 1st and 2nd Ajima-Malfatti points (1ST AND 2ND Ajima-Malfatti Points)
- The Apollonius point — not to be confused with the Apollonius points
- The Bailey points (Bailey Point)
- The Hofstadter points (Hofstadter Points)[20]
- The congruent isoscelizers point (Congruent Isoscelizers Point)[21]
- The 1st and 2nd Morley centers, associated with the Morley triangle (1ST AND 2ND Morley Centers)[22]
- The Parry point (Parry Point)[23]
- The isoperimetric point and the equal detour point (Isoperimetric Point and Equal Detour Point)[24]
- The equal parallelians points (Equal Parallelians Point)[25]
- The Schiffler point (Schiffler Point)[26]
- The Exeter point
- The point of division of a triangle into three triangles with three equal radii of the three inscribed circles
See also
- Notable lines of a triangle
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