Important Lines and Points related to a Triangle

In this article, we are going to learn about some important lines and points that are related to a triangle. You will encounter many questions in aptitude exams from this section.

If we drop a line from any vertex of a triangle to the opposite side (or extended opposite side), then that line is called a Cevian. Geometry There are various types of cevians in a triangle: Medians, Altitudes, Angle Bisectors.

We will study about all these, along with some other important lines like Perpendicular Bisectors.

Medians

A median is a straight line that joins a vertex of a triangle to the mid-point of the opposite side. Geometry If AD = DB, then CD must be a median.

Properties of Medians

Property 1: Position of Medians

As a triangle has three vertices, it obviously will have three medians. And all of the three medians will always lie inside the triangle. This is true for any triangle.

Property 2: Length of Medians

  • In equilateral triangle, all the three medians have the same length.

  • In scalene triangle, all the three medians are unequal.

  • In Isosceles triangle, two medians are equal, while the third one is of a different length. Medians drawn from vertex of two equal angles are equal. While, the median drawn from vertex of the unequal angle is of a different length, is perpendicular to the side and bisects the vertex angle. Geometry

  • In right-angled triangle, the median drawn from vertex of right angle to hypotenuse is equal to half of the hypotenuse. Geometry AD = $\frac{1}{2}$ BC

The other way round is also true. That is, if a median is equal to half of its corresponding side, then it means that the triangle must be a right-angled triangle.

  • Perimeter of a triangle > Sum of the three medians of a triangle

  • Sum of any two sides of a triangle > Twice the median drawn to the third side. Geometry CA + CB > 2 CD

Now, let’s see some more properties related to medians.

Property 3

Medians divide the triangle into six small triangles. Each of these small triangles have equal area. Geometry

Property 4

A line segment (AQ) joining any vertex and midpoint of a median (P), if extended, divides the third side (CB) into a ratio of 1 : 2. Geometry

If CD is median, and P is midpoint of CD then:

  • CQ : QB = 1 : 2 OR
  • CQ = $\frac{1}{3}$ CB

Centroid

Centroid of a triangle is a point where all of its three medians intersect. Geometry

The three medians of any triangle are always concurrent. That is, they always have a common point of intersection.

Property 1

Centroid divides the median in a ratio of 2 : 1. Geometry For example, in the above figure OC : OD = 2 : 1

Property 2

If we form a triangle by joining the centroid, and the midpoints of any two sides, then its area will be $\frac{1}{12}$th of the area of the original triangle. Geometry Area of ∆OPQ = $\frac{1}{12}$ × Area of ∆ABC



Altitudes

An altitude of a triangle, is a perpendicular line segment drawn from any vertex to the opposite side. So, it’s basically the height of a triangle. Geometry In the above figure, CD is the altitude with respect to the side AB.

Obviously, an altitude is going to be the shortest distance from a vertex to its opposite side.

Properties of Altitudes

Property 1: Position of Altitudes

  • In any acute angled triangle, all the three altitudes lie inside the triangle.

  • In a right angled triangle, the two sides containing right angle are also the altitudes. While, the altitude from vertex of right angle to hypotenuse will lie inside the triangle. Geometry In the above figure, CA, BA & AD are altitudes on the sides AB, AC & BC respectively.

  • In obtuse angled triangle, two altitudes from acute angles will lie outside of the triangle. While the altitude from the obtuse angle will lie inside of the triangle. Geometry In the above figure, AP, BQ and CR are altitudes on the sides BC, AC & AB respectively.

Property 2: Length of Altitudes

  • The longest side has the least corresponding altitude. Similarly, the shortest side has the longest corresponding altitude.

  • In equilateral triangle, altitude and median are the same. And so, just as was the case with medians, all the three altitudes are equal in length. Geometry

  • In scalene triangle, all the three altitudes are unequal.

  • In Isosceles triangle, two altitudes are equal, while the third one is of a different length. Altitudes/Perpendiculars drawn from vertex of two equal angles (to opposite equal sides) are equal. While, the altitude/perpendicular drawn from vertex of the unequal angle (to opposite unequal side) is of a different length, is median to the side and bisects the vertex angle (i.e. it is altitude, median and angular bisector). Geometry If CA = CB then AP = BQ
    CR is not only the altitude, but also the median and angular bisector of ∠ACB.

Orthocentre

Orthocentre of a triangle is a point where all of its three altitudes intersect. Geometry

The three altitudes of any triangle are always concurrent. That is, they always have a common point of intersection.

Property 1: Position of Orthocentre

  • In any acute angled triangle, orthocentre will lie inside the triangle. Geometry

  • In any right angled triangle, orthocentre is the vertex of the right angle. Geometry

  • In any obtuse angled triangle, orthocentre will lie outside the triangle. Geometry

Property 2

  • In any equilateral triangle, the centroid and orthocentre are the same point.

  • In isosceles triangle, vertex, centroid and orthocentre are collinear points.

  • In scalene triangle, vertex, centroid and orthocentre are three non-collinear points.

Sum of three altitudes of a triangle < Sum of three sides of the triangle

Property 3

We already know that, in any isosceles triangle centroid and orthocentre are different, but collinear points.

Moreover, in any isosceles triangle, both of these points lie on the altitude and median drawn from the vertex of the unequal angle to the opposite unequal side. Geometry

The other way round is also true. If in a triangle, we find that the line joining the centroid and orthocentre is the altitude of any side and also bisects it (as a median does), then that triangle must be an isosceles triangle.

Property 4

Angle made by any side of a triangle on ortho-centre is supplementary of its opposite angle. Geometry So, ∠AOB + ∠ACB = 180°



Angle Bisectors

An angle bisector of a triangle, is a line segment joining a vertex to the opposite side, such that it bisects the vertex angle.
Geometry In the above figure, CD is the angle bisector of the angle ∠ACB.

Properties of Angle Bisectors

Property 1: Position of Angle Bisectors

As a triangle has three vertices, it obviously will have three angle bisectors. And all of the three angle bisectors will always lie inside the triangle (just like medians). This is true for any triangle.

Property 2: Length of Angle Bisectors

  • In any equilateral triangle, all the three angle bisectors have the same length.
In any equilateral triangle, median, altitude and angle bisector are the same line segment.

  • In any scalene triangle, all the three angle bisectors are unequal in length.

  • In any isosceles triangle, two angle bisectors are equal, while the third one is of a different length. Angle bisectors drawn from vertex of two equal angles are equal. While, the angle bisector drawn from vertex of the unequal angle is of a different length, is altitude, as well as median to the opposite side. Geometry In the above figure, ∆ABC is an isosceles triangle, as ∠A = ∠B. AP, BQ & CR are angle bisectors, and AP = BQ.

Property 3

Angular bisector of a triangle divides the opposite side in the ratio of the other two sides. Geometry If in the above figure CD is an angular bisector, then $\frac{AD}{BD}$ = $\frac{AC}{BC}$

In other words, we can say that if a line passing through a vertex of a triangle divides the opposite side in the ratio of the other two sides, then the line must be the angular bisecor of the angle at the vertex.
Similarly, the external angular bisector of an angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle. Geometry If in the above figure CD is an external angular bisector, then $\frac{AD}{BD}$ = $\frac{AC}{BC}$

Property 4: Angle made by two Angular Bisectors

Angular Bisectors of interior angles

The angle made by angular bisectors of any two angles = 90° + $\frac{1}{2}$ × Third angle. Geometry ∠ADB = 90° + $\frac{1}{2}$ ∠ACB

In other words, we can say that the angle made by any side of a triangle on incentre is equal to 90° + $\frac{1}{2}$ × Opposite angle

We will study about incentre soon.

Angular Bisectors of exterior angles

The angle made by angular bisectors of any two exterior angles = 90° - $\frac{1}{2}$ × Third angle. Geometry ∠ADB = 90° - $\frac{1}{2}$ ∠ACB

Angular Bisectors of interior angle and exterior angle

The angle made by angular bisectors of one interior and one exterior angle = $\frac{1}{2}$ × Third angle. Geometry ∠ADB = $\frac{1}{2}$ ∠ACB

Property 5

Angle made by altitude and angle bisector on a vertex = Half of the difference between other two opposite angles Geometry In the above figure, CD is the altitude (CD ⊥ AB) and CE is the angular bisector of ∠C.
So, ∠DCE = $\frac{1}{2}$ (∠A – ∠B)

Incentre

Incentre of a triangle is a point where all of its three internal bisectors intersect. Geometry

The three internal bisectors of any triangle are always concurrent. That is, they always have a common point of intersection.

Property 1: Position of Incentre

In any triangle, incentre will always lie inside the triangle.

Property 2

Incentre of a triangle is equidistant from all the three sides of that triangle. Geometry In the above figure, IP = IQ = IR

Property 3

  • In any equilateral triangle, centroid, orthocentre, and incentre are the same point.

  • In any isosceles triangle, centroid, orthocentre and incentre are three different collinear points.

  • In any scalene triangle, centroid, orthocentre and incentre are three different non-collinear points.

Property 4

We already know that, in any isosceles triangle centroid, orthocentre and incentre are different, but collinear points.

Moreover, in any isosceles triangle, all of these three points lie on the angle-bisector, altitude, and median drawn from the vertex of the unequal angle to the opposite unequal side.

Incircle

Incircle is a circle inside a triangle, such that:

  • it touches all the three sides of that triangle and
  • its centre is the incentre of that triangle
Geometry

Inradius formulae

Inradius Formula 1

Inradius of any triangle = $\frac{Area \hspace{1ex} of \hspace{1ex} Triangle}{Semi \hspace{1ex} Perimeter \hspace{1ex} of \hspace{1ex} Triangle}$

Inradius Formula 2
Geometry Inradius of right-angled triangle = $\frac{Base + Height - Hypotenuse}{2}$

Excentre

Excentre of a triangle is a point where one of its internal bisectors and two of its external bisectors intersect, i.e. it is the intersection point of the internal bisector of one angle and the external bisectors of the other two angles. Geometry

Excircle

An excircle or escribed circle of a triangle is tangent to one of its sides and also tangent to the extensions of the other two sides. It always lies outside the triangle.

Every triangle has three distinct excircles, each tangent to one of the sides of the triangle.

  • Excircles of a triangle are always larger than its incircle.

  • The largest excircle of a triangle is tangent to its longest side, while the smallest excircle is tangent to its shortest side.

Exradii

The radii of the excircles are called the exradii. As we have 3 excircles in a triangle, we have 3 exradii too. Geometry

In the triangle shown above, if a, b & c are the sides of the triangle ABC, s is the semi-perimeter of the triangle ABC, and ∆ is the area of the triangle ABC, then:

$r_1$ = $\frac{∆}{s - a}$

$r_2$ = $\frac{∆}{s - b}$

$r_3$ = $\frac{∆}{s - c}$



Perpendicular Bisectors

Perpendicular Bisector of a side of a triangle is a line that passes through the mid-point of that side and is also perpendicular to it. Geometry

Unlike medians, altitudes and angle bisectors, it is not necessary for a perpendicular bisector to pass through any vertex.

Properties of Perpendicular Bisectors

Property 1: Length of Perpendicular Bisectors

  • In equilateral triangle, all the three perpendicular bisectors have the same length.
In any equilateral triangle, median, altitude, angle bisector and perpendicular bisector are the same line segment.

  • In any scalene triangle, all the three perpendicular bisectors are unequal in length. None of them pass through the vertices.

  • In any isosceles triangle, two perpendicular bisectors are equal, while the third one is of a different length. Perpendicular bisectors drawn on equal sides are equal. While, the perpendicular bisector drawn on the unequal side is of a different length, is median, altitude, as well as angle bisector. Geometry In the figure given above, PP’ = QQ’

Perpendicular bisectors drawn on equal sides of an isosceles triangle do not pass through the vertices.

Circumcentre

Circumcentre of a triangle is a point where the three perpendicular bisectors of its sides intersect. Geometry

In other words, the line segment joining circumcentre and midpoint of any side of the triangle must be perpendicular on that side and vice-versa.
The three perpendicular bisectors of any triangle are always concurrent. That is, they always have a common point of intersection.

Property 1: Position of Circumcentre

  • In any acute angled triangle, circumcentre will always lie inside the triangle.
  • In any right angled triangle, circumcentre will always be at the midpoint of hypotenuse.
  • In any obtuse angled triangle, circumcentre will always lie outside of the triangle.

Property 2

Circumcentre of a triangle is equidistant from all the three vertices of that triangle. Geometry In the above figure, DA = DB = DC

Property 3

  • In any equilateral triangle, centroid, orthocentre, incentre and circumcentre are the same point.

  • In any isosceles triangle, centroid, orthocentre, incentre and circumcentre are four different collinear points.

  • In any scalene triangle, centroid, orthocentre, incentre and circumcentre are four different non-collinear points.

Property 4

We already know that, in any isosceles triangle centroid, orthocentre, incentre and circumcentre are different, but collinear points.

Moreover, in any isosceles triangle, all of these four points lie on the line segment joining vertex and midpoint of unequal side.

Property 5

Angle made by any side of a triangle on circumcentre is twice of the opposite angle. Geometry In the above figure, ∠ADC = 2 ∠ABC

Angle formed by a chord (AB) of a circle on major arc is half the angle formed by it on the centre of that circle.

Circumcircle

Circumcircle is a circle that passes through all the three vertices of a triangle and its centre is the circumcentre of that triangle. Geometry

Circumradius formulae

Circumradius formula 1

Length of circumradius = $\frac{Product \hspace{1ex} of \hspace{1ex} sides}{4 × Area \hspace{1ex} of \hspace{1ex} triangle}$

Circumradius formula 2

In any right angled triangle, the hypotenuse is the diameter of its circumcircle.

Circumradius formula 3

In any equilateral triangle, which has sides of length ‘a’ each:

In-radius = $\frac{a}{2\sqrt{3}}$ and Circumradius = $\frac{a}{\sqrt{3}}$
So, Inradius : circumradius = 1 : 2
Hence, Area of incircle : Area of circumcircle= 1 : 4

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Last updated on Jan 18, 2022

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