**Segments in a flat triangle** are those that unite the vertex of a triangle with the opposite side (or its extension), as part of a line or ray, in different ways and in various cases arise with their own name; in certain classes of triangles (isosceles, regular) and two or more coincide. In addition, they are other segments traced by particular points on the sides (perpendicular bisector).

## Summary

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- 1 Initial nomenclature
- 2 Height
- 3 Bisector
- 4 Midline
- 5 Medium
- 6 Perpendicular
- 7 Ceviana
- 8 Simediana
- 9 References
- 10 Sources

## Initial nomenclature

a, b, c, their opposite angles A, B, C.

semiperimeter p = (a + b + c) ÷ 2

S = area of the triangle region

## Height

It is the perpendicular segment drawn to the opposite or the extension of said side. Let ΔABC be the height AH _{1} , BH _{2} , CH _{3} , denoted h _{a} , h _{b} , h _{a} . H _{i} , i = 1,2,3 is called foot height. All three intersect at a point called an **orthocenter** .

Orthotocenter **O** of Δ Acute Angle

In a ut scalene acutangle, all three feet tall are on the opposite side; the orthocenter is inside the Δ. In an obtuse scalene Δ the feet of the heights drawn from the vertex of acute angles are on the extensions of their opposite sides and the foot of the obtuse angle is on the opposite side; the orthocenter is on the outside of the figure. In a Δ scalene rectangle the heights that start from the acute angles coincide with the legs, the height of the right angle has its foot on the hypotenuse; its orthocenter coincides with the vertex of the right angle.

In an isosceles triangle , heights of equal sides have equal measure; if it is an acute angle, the orthocenter inside the figure and if it is obtuse on the outside.

In a regular triangle the three heights have the same length, the orthocenter is the interior point of the figure.

Formula

h _{a} = 2 [p (pa) (pb) (pc)] ÷ a

h _{b} = 2 [p (pa) (pb) (pc)] ÷ b

h _{c} = 2 [p (pa) (pb) (pc)] ÷ c

## Bisector

It is a line that divides the interior angle of a triangle into two equal parts. The bisectors meet at the interior point, called the **incenter** , which is nothing but the center of the inscribed circumference of radius r. In the ΔABC, let the angles α with vertex at A, β at B and γ at C.

Incenter **I** of the ΔABC

Expressions for bisectors

l _{α} = 2 [bcp (pa)] ^{0.5} : (b + c)

l _{β} = 2 [acp (pb)] ^{0.5} : (a + c)

l _{γ} = 2 [abp (pc)] ^{0.5} : (b + a)

Parts on one side with the other sides.

Let triangle ABC be the bisector l _{α} , which starts from A, and cuts a = BC at point D, with BD = m, DC = n, the sides that converge in A, AC = b, AB = c, so

m: n = b: c

According to triangular class

- In an isosceles triangle the bisectors of the angles at the base are the same; and the bisector of the angle at the vertex (formed by the equal sides) coincides with the median, perpendicular bisector and height.
- In a regular triangle all three bisectors are equal (the same length). Then the bisector coincides with the median, perpendicular bisector, and height.

## Middle line

In a triangle, the segment that joins the midpoints of any pair of sides is called the **midline** ; Let L be the midpoint of BC; M, from AC and N, from AB.

Properties

Each median line is // to the side that does not contain the midpoints, plus its length is half that of the parallel side.

MN // AC length of MN = a / 2

LN // AC length of LN = b / 2

LM // AB length of LM = c / 2

The ΔLMN, with vertices at the midpoints is similar to ΔABC and its area is 1/4 of the area of the container triangle.

## Median

The segment **median** of a triangle is called a vertex with the midpoint of the opposite side of the triangle. The medians of a triangle intersect at an interior point called a **barycenter** (center of gravity).

Medians of a triangle

The barycenter cuts each median into two segments , one half of the other:

Formulas

m _{a} = 0.5 (2b ^{2 + 2c }^{2} -a ^{2} ) ^{0.5}

m _{b} = 0.5 (2a ^{2 + 2c }^{2} -b ^{2} ) ^{0.5}

m _{c} = 0.5 (2b ^{2 + 2a }^{2} -c ^{2} ) ^{0.5}

Properties

The centroid divides a median at the ratio 2: 1, counting from the vertex of the angle

The median is less than the half-sum of the sides that form the vertex of the triangle from which the median starts.

In an isosceles triangle the medians that come out of the vertices of the angles at the base are the same.

In a regular triangle the three medians are equal and their length m, using the length a of the common side, is:

M = a × (3) ^{0.5} : 2.

## Mediatrix

The **perpendicular** line at the midpoint of either side is called the **perpendicular bisector** of a triangle.

Properties

The perpendicular bisectors of a triangle meet at the same point, which has the same distance from the three vertices. The common point is called the **circumcenter** and is the center of the circumscribed circumference that passes the vertices of the triangle, whose radius R is the distance from the circumcenter to any vertex.

The area of the ΔABC is S = abc: 4R, where a, b and c are the sides and r are the radius of the circumscribed circumference.

(27) ^{0.5} r ^{2} ≤ S _{Δ} ≤ (27) ^{0.5} R ^{2} ÷ 4

2 r ≤ R; r radius of the circle inscribed in the triangle and R, that of the circumscribed circle.

## Ceviana

Any line drawn from the vertex of a triangle and passes through the opposite side or through the extension of the opposite side is called **ceviana** . In such a way that the median, the bisector, the height considered as straight can be understood as medians ^{[1]} .

Ceva’s theorem

When we draw the cevianas AL, BM and CN through the vertices of a ΔABC, they intersect at a single point if and only if

AN / NB × BL / LC × CM / MA = 1 ^{[2]}

## Simediana

It is called **simediana** to the line of a triangle which is symmetrical relative to the median bisector ^{[3]}

Properties

- The Simedian divides the respective side of the triangle in direct proportion to the squares of the other two sides.
- The simedian is the locus of all points for which the distances to the adjacent sides are directly proportional to the lengths of such sides.
- The common point of the three simedianas coincides with the
**Lemoine point**, this is the point whose distances to the three sides are proportional to these^{[4]}

Formula

s _{a} = bc [2 (b ^{2} + c ^{2} ) – a ^{2} ] ^{0.5} ÷ (b ^{2} + c ^{2} ) ^{[5]}