The Incenter, Incircle and Inradius of a triangle

The incenter of a triangle is the point of intersection of the interior angle bisectors of that triangle.

In the following triangle \(▵ABC\),
\(AI\), \(BI\) and \(CI\) are the interior angle bisectors of the triangle.
They intersect at the point \(I\), which is the incenter of \(△ABC\).

The incenter point of a triangle always lies inside the triangle, and is equidistant from all the sides of that triangle.

Incenter of a triangle

What are an incircle and an inradius?

The incircle of a triangle is a largest circle which can be contained in that triangle. The center of an incircle is known as the incenter, while radius of the incircle is called the inradius.
An incircle is also known as the inscribed circle.

What is an angle bisector?

An angle bisector is a line segment that bisects an angle in two equal angles.

In the above, \(∠ABC = 80° \)
When it is bisected by the segment \(BD\),
produces two equal angles of \(40°\).
So, \(BD\) is the angle bisector of \(∠ABC \).

Properties of an incenter

• ✓ Incenter is the point of intersection of the interior angle bisectors.
• ✓ It is the centre of the incircle of the triangle.
• ✓ An incenter is equidistant from all the sides of the triangle .
• ✓ An incenter point always lies inside the triangle.
• ✓ In an equilateral triangle, the centroid, the orthocenter,
•  the circumcenter and the incenter coincide at the same point.

\(AI\), \(BI\) and \(CI\) are the angle bisectors.
\(I\) is the incenter of the triangle.
The circle is the incircle of the triangle.
\(IX\), \(IY\) and \(IZ\) are the inradii.

\(\therefore IX=IY=IZ\)
In right-angled triangles \(▵AZI\) and \(▵AXI, \;IX=IZ\),
hypotenuse \(\;AI\) is common to both.
So, by RHS rule,\(\quad▵AZI ≅ ▵AXI\),
and by CPCT rule, \(\quad AX = AZ\)
Similarly,\(\quad▵BXI ≅ ▵BYI, \;BX=BY\)
and\(\quad▵CYI ≅ ▵CZI, \;CY=CZ\)

Incenter of an Acute triangle

\(AI\), \(BI\) and \(CI\) are the angle bisectors.
\(I\) is the incenter of the triangle.
The circle is the incircle of the triangle.
\(IX\), \(IY\) and \(IZ\) are the inradii.

Incenter of an Obtuse triangle

Incenter of a Right-angled triangle

Coordinates of incenter

If coordinates of the vertices \(A\), \(B\) and \(C\) are \( (-18, 9) \), \( (11, 9) \) and \( (-9, 33) \) respectively, and
\(c = \) length of \(AB\),
\(a = \) length of \(BC\) and
\(b = \) length of \(CA\),
then the coordinates of the incenter \(I(x, y)\) can be calculated with the following formula:

Coordinates of \(I(x, y) \)
\[ =\left(\frac{ax_1+bx_2+cx_3}{a+b+c}\;, \frac{ay_1+by_2+cy_3}{a+b+c}\right) \] Using distance formula \(\; \sqrt{(y_2-y_1)^2 + (x_2-x_1)^2} \) ,
first we calculate the length of the sides of the triangle.

\(\; \scriptsize c=AB= \sqrt{(11 - (-18))^2 + (9 - 9)^2} = \sqrt{29^2 + 0^2} = 29\)
\(\; \scriptsize a=BC= \sqrt{(-9 - 11)^2 + (33 - 9)^2} = \sqrt{(-20)^2 + 24^2} = 31.24 \)
\(\; \scriptsize b=CA= \sqrt{(-18 - (-9))^2 + (9 - 33)^2} = \sqrt{(-9)^2 + (-24)^2} = 25.63 \)

\(\therefore\;\) Coordinates of \(I(x, y) \)
\begin{align} & = \scriptsize \left(\frac{31.24\times(-18)+25.63\times 11+29\times (-9)}{31.24+25.63+29},\; \frac{31.24\times 9+25.63\times 9+29\times 33}{31.24+25.63+29}\right) \\\\ & = \scriptsize \left(\frac{-562.32 + 281.93- 261}{85.87},\; \frac{281.16 + 230.67 + 957}{85.87}\right) \\\\ & = \left(\frac{-541.39}{85.87},\; \frac{1468.83}{85.87}\right) \\\\ & = (-6.3,\; 17.1) \\\\ \end{align}

\(\; c=AB= \sqrt{(11 - (-18))^2 + (9 - 9)^2} = \sqrt{29^2 + 0^2} = 29\)
\(\; a=BC= \sqrt{(-9 - 11)^2 + (33 - 9)^2} = \sqrt{(-20)^2 + 24^2} = 31.24 \)
\(\; b=Ca= \sqrt{(-18 - (-9))^2 + (9 - 33)^2} = \sqrt{(-9)^2 + (-24)^2} = 25.63 \)

\(\therefore\;\) Coordinates of \(I(x, y) \)
\begin{align} & = \left(\frac{31.24\times(-18)+25.63\times 11+29\times (-9)}{31.24+25.63+29},\; \frac{31.24\times 9+25.63\times 9+29\times 33}{31.24+25.63+29}\right) \\\\ & = \left(\frac{-562.32 + 281.93- 261}{85.87},\; \frac{281.16 + 230.67 + 957}{85.87}\right) \\\\ & = \left(\frac{-541.39}{85.87},\; \frac{1468.83}{85.87}\right) \\\\ & = (-6.3,\; 17.1) \\\\ \end{align}

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