Circumcentre and Incentre of a Triangle

We will discuss circumcentre and incentre of a triangle.

In general, the incentre and the circumcentre of a triangle are two distinct points.

Incentre and Circumcentre of a Triangle

Here in the triangle XYZ, the incentre is at P and the circumcentre is at O. 

A special case: an equilateral triangle, the bisector of the opposite side, so it is also a median.

In the ∆XYZ, XP, YQ and ZR are the bisectors of ∠YXZ, ∠XYZ and ∠YZX respectively; they are also the perpendicular bisectors of YZ, ZX and XY respectively; they are also the medians of the triangle. So, their point of intersection, G, is the incentre, circumcentre as well as the centroid of the triangle. So, in an equilateral triangle, these three points are coincident.

Incentre, Circumcentre and Centroid of an Equilateral Triangle

If XY = YZ = ZX = 2a then in ∆XYP, YP = a and XP = \(\sqrt{3}\)a.

Now, XG = \(\frac{}{}\) = \(\frac{2}{3}\)XP = \(\frac{2\sqrt{3}a}{3}\), and GP = \(\frac{1}{3}\)XP = \(\frac{\sqrt{3}a}{3}\).

Therefore, radius of the circumcircle is XG = \(\frac{2\sqrt{3}a}{3}\) = \(\frac{2a}{\sqrt{3}}\) = \(\frac{Any side of the equilateral triangle}{\sqrt{3}}\).

Incentre, Circumcentre & Centroid of an Equilateral Triangle

The radius of the incircle = GP = \(\frac{a}{\sqrt{3}}\) = \(\frac{2a}{2\sqrt{3}}\) = \(\frac{Any side of the equilateral triangle}{2\sqrt{3}}\).

Therefore, radius of the circumcircle of an equilateral triangle = 2 × (Radius of the incircle).






10th Grade Math

From Circumcentre and Incentre of a Triangle to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Thousandths Place in Decimals | Decimal Place Value | Decimal Numbers

    Jul 20, 24 03:45 PM

    Thousandths Place in Decimals
    When we write a decimal number with three places, we are representing the thousandths place. Each part in the given figure represents one-thousandth of the whole. It is written as 1/1000. In the decim…

    Read More

  2. Hundredths Place in Decimals | Decimal Place Value | Decimal Number

    Jul 20, 24 02:30 PM

    Hundredths Place in Decimals
    When we write a decimal number with two places, we are representing the hundredths place. Let us take plane sheet which represents one whole. Now, we divide the sheet into 100 equal parts. Each part r…

    Read More

  3. Tenths Place in Decimals | Decimal Place Value | Decimal Numbers

    Jul 20, 24 12:03 PM

    Tenth Place in Decimals
    The first place after the decimal point is tenths place which represents how many tenths are there in a number. Let us take a plane sheet which represents one whole. Now, divide the sheet into ten equ…

    Read More

  4. Representing Decimals on Number Line | Concept on Formation of Decimal

    Jul 20, 24 10:38 AM

    Representing decimals on number line shows the intervals between two integers which will help us to increase the basic concept on formation of decimal numbers.

    Read More

  5. Decimal Place Value Chart |Tenths Place |Hundredths Place |Thousandths

    Jul 20, 24 01:11 AM

    Decimal place value chart
    Decimal place value chart are discussed here: The first place after the decimal is got by dividing the number by 10; it is called the tenths place.

    Read More