Excenter of a triangle

In this article, we will cover the concepts of Excircle. This category falls under the broader category of Coordinate Geometry, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. A total of six questions have been asked on JEE MAINS( 2013 to 2023) from this topic.

Excenters of Triangle: Definition

An excircle or escribed circle of a triangle is a circle that lies outside the triangle such that it is tangent to one of the triangle’s sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, one on each side. The center of an excircle is the intersection of the internal bisector of one angle of the triangle and the external bisectors of the other two angles. The center of the excircle is called the excentre relative to the vertex opposite the side tangent to the circle.

Excentre of Triangle Formula

An excenter is a point at which the line bisecting one interior angle meets the bisectors of the two exterior angles on the opposite side.

Let $A\left(x_1, y_1\right) B\left(x_2, y_2\right)$ and $C\left(x_3, y_3\right)$ be the vertices of triangle ABC such that BC = a, CA = b, and AB = c.

The circle that touches the side BC, AB produced, and AC produced is called the escribed circle opposite the angle A.

The bisectors of the external angle B and C meet at a point I₁ which is the centre of the escribed circle opposite the angle A.

If I₁ is the point of intersection of the internal bisector of ∠BAC and the external bisector of ∠ABC and ∠ACB then,

Coordinates of I₁ , I₂ and I₃ is given by

$\begin{aligned} & \mathrm{I}_1 \equiv\left(\frac{-a x_1+b x_2+c x_3}{-a+b+c}, \frac{-a y_1+b y_2+c y_3}{-a+b+c}\right) \\ & I_2 \equiv\left(\frac{a x_1-b x_2+c x_3}{a-b+c}, \frac{a y_1-b y_2+c y_3}{a-b+c}\right) \\ & I_3 \equiv\left(\frac{a x_1+b x_2-c x_3}{a+b-c}, \frac{a y_1+b y_2-c y_3}{a+b-c}\right)\end{aligned}$

Properties of Excentre

1) Exacentres always lie outside the Triangles.

2) The excentres of an equilateral triangle are equidistant from each other and this distance is twice the side of the triangle.

3) Each excenter is the center of an excircle tangent to one side of the triangle and the extensions of the other two sides.

4) The excenter is the intersection point of the internal angle bisector of one vertex and the external angle bisectors of the other two vertices.

5) The three excenters and the orthocenter of the triangle are collinear.

6) The incenter and the three excenters form an orthocentric system, with the incenter as the orthocenter of the triangle formed by the excenters.

Problems Based on Excentre

Example 1: If $\triangle A B C$ is an equilateral triangle of side a then find the minimum distance between two excentres.

Solution:

Let in $\triangle A B C \mathrm{~A}(0,0), B\left(\frac{a}{2}, \frac{\sqrt{3} a}{2}\right), C(a, 0)$
$
\begin{aligned}
& \mathrm{I}_1 \equiv\left(\frac{-\mathrm{ax} 1+\mathrm{bx} \mathrm{x}_2+\mathrm{cx} \mathrm{x}_3}{-\mathrm{a}+\mathrm{b}+\mathrm{c}}, \frac{-\mathrm{ay}_1+\mathrm{by}_2+\mathrm{cy} y_3}{-\mathrm{a}+\mathrm{b}+\mathrm{c}}\right) \\
& \mathrm{I}_2 \equiv\left(\frac{\mathrm{ax} 1-\mathrm{bx} 2+\mathrm{cx} 3}{\mathrm{a}-\mathrm{b}+\mathrm{c}}, \frac{\mathrm{ay} 1{ }_1-\mathrm{by}_2+c y_3}{\mathrm{a}-\mathrm{b}+\mathrm{c}}\right) \\
& \mathrm{I}_3 \equiv\left(\frac{\mathrm{ax} 1+\mathrm{bx} 2-\mathrm{cx} 3}{\mathrm{a}+\mathrm{b}-\mathrm{c}}, \frac{\mathrm{ay}_1+\mathrm{by} y_2-c y_3}{\mathrm{a}+\mathrm{b}-\mathrm{c}}\right) \\
& I_1=\left(\frac{-a \cdot 0+a \cdot a+a \cdot \frac{a}{2}}{-a+a+a}, \frac{-a \cdot 0+a \cdot 0+a \cdot \frac{\sqrt{3} a}{2}}{-a+a+a}\right) \\
& I_1=\left(\frac{3 a}{2} \cdot \frac{\sqrt{3} a}{2}\right) \\
&
\end{aligned}
$

Hence the answer is $2 a$

Example 2: ABC is an equilateral triangle such that the vertices B and C lie on two parallel lines at a distance 6. If A lies between the parallel lines at a distance 4 from one of them then the length of a side of the equilateral triangle is

Solution: From the choice of the axis $\mathrm{A}=(0,0), \mathrm{B}=(2 \cot \theta, 2), \mathrm{C}=\left(4 \cot \left(60^{\circ}-\theta\right),-4\right)$

Now (side of equilateral triangle)²

$\begin{aligned} & =4 \cot ^2 \theta+4=16 \cot ^2(60-\theta)+16 \\ & \Rightarrow 4 \operatorname{cosec}^2 \theta=16 \operatorname{cosc}^2(60-\theta) \\ & \Rightarrow \tan \theta=\frac{\sqrt{3}}{5}\end{aligned}$
Hence the required length is $=2 \operatorname{cosec} \theta=4 \sqrt{\frac{7}{3}}$.

Hence, the correct answer is $4 \sqrt{\frac{7}{3}}$

Example 3: $\mathrm{P}(\mathrm{x}, \mathrm{y})$ is called a good point if $x, y \in N$. The total number of good points lying inside the quadrilateral formed by the line $2 \mathrm{x}+\mathrm{y}=2, \mathrm{x}=0, \mathrm{y}=0$ and $\mathrm{x}+\mathrm{y}=5$, is equal

Solution: The adjacent figure indicates that there are exactly six good points inside the quadrilateral ABCD.

Hence, the correct answer is 6

Example 4: If the point $\mathrm{P}\left(\mathrm{a}, \mathrm{a}^2\right)$ lies completely inside the triangle formed by the lines $\mathrm{x}=0, \mathrm{y}=0$ and $\mathrm{x}+\mathrm{y}=2$,then exclusive range of $\text { 'a' }$ is

Solution: Clearly $a \in R^{+}$. Also $a^2+a-2<0$


$\begin{aligned} & \Rightarrow(\mathrm{a}+2)(\mathrm{a}-1)<0 \\ & \Rightarrow-2<\mathrm{a}<1 \\ & \Rightarrow \mathrm{a} \in(0,1)\end{aligned}$
Hence, the correct answer is $a \in(0,1)$

Example 5: If one vertex of an equilateral triangle of side 2 is the origin and another vertex lies on the line $x=\sqrt{3} y$ then the third vertex can be

Solution: Clearly the third vertex will lie on the $y_{-}$axis. Hence the points are $(0,2)$ or $(0,-2)$



Hence, the correct answer is (0,2)

Summary

Excircles are geometric elements related to triangles. It provides the relationship between sides, angles, and circles in triangle geometry. Excircles not only contribute to geometric properties and triangle configurations but also play crucial roles in various theorems and proofs. Understanding excircles helps us in theoretical as well as practical knowledge.