Solution of Triangles: Circles connected with a triangle - incircle
Solution of Triangles: Circles connected with a triangle - incircle
The study of triangles and the relationships between their sides and angles is a fundamental aspect of trigonometry. One interesting concept within this field is the incircle of a triangle. An incircle is a circle that is tangent to each side of the triangle, and it is located within the triangle. The center of the incircle is called the incenter, and it is the point where the angle bisectors of the triangle intersect.
Properties of the Incircle
Before diving into the formulas and examples, let's summarize some key properties of the incircle:
| Property | Description |
|---|---|
| Incenter | The point of intersection of the angle bisectors of the triangle. |
| Radius | The radius of the incircle is equal to the area of the triangle divided by the semiperimeter. |
| Tangency Points | The points where the incircle touches the sides of the triangle. |
| Semiperimeter | Half the sum of the sides of the triangle. |
Formulas Related to the Incircle
The following formulas are essential when dealing with the incircle of a triangle:
Semiperimeter (s): $$ s = \frac{a + b + c}{2} $$ where $a$, $b$, and $c$ are the lengths of the sides of the triangle.
Area (A): $$ A = \sqrt{s(s-a)(s-b)(s-c)} $$ This is known as Heron's formula.
Radius of Incircle (r): $$ r = \frac{A}{s} $$ This formula derives from the fact that the area of the triangle can also be expressed as the semiperimeter multiplied by the radius of the incircle.
Length of Tangents from a Point to the Incircle (l): If a point is at a distance $d$ from the incenter, then the length of the tangents from the point to the incircle (assuming they exist) is given by: $$ l = \sqrt{d^2 - r^2} $$
Examples
Let's go through a couple of examples to understand how these concepts and formulas are applied.
Example 1: Finding the Radius of the Incircle
Given a triangle with sides $a = 7$ units, $b = 8$ units, and $c = 9$ units, find the radius of the incircle.
Solution:
Calculate the semiperimeter: $$ s = \frac{7 + 8 + 9}{2} = 12 \text{ units} $$
Use Heron's formula to find the area: $$ A = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \cdot 5 \cdot 4 \cdot 3} = \sqrt{720} = 6\sqrt{20} \text{ units}^2 $$
Calculate the radius of the incircle: $$ r = \frac{A}{s} = \frac{6\sqrt{20}}{12} = \frac{\sqrt{20}}{2} = \sqrt{5} \text{ units} $$
The radius of the incircle is $\sqrt{5}$ units.
Example 2: Finding the Length of Tangents to the Incircle
Given the same triangle as in Example 1, suppose there is a point $P$ inside the triangle such that it is $5$ units away from the incenter. Find the length of the tangents from $P$ to the incircle.
Solution:
We already know the radius of the incircle from Example 1, which is $\sqrt{5}$ units.
Use the formula for the length of tangents: $$ l = \sqrt{d^2 - r^2} = \sqrt{5^2 - (\sqrt{5})^2} = \sqrt{25 - 5} = \sqrt{20} \text{ units} $$
The length of each tangent from point $P$ to the incircle is $\sqrt{20}$ units.
Conclusion
The incircle of a triangle is a circle that fits snugly inside the triangle, touching all three sides. The center of this circle, the incenter, is equidistant from all sides of the triangle and is found at the intersection of the angle bisectors. The radius of the incircle can be found using the area and semiperimeter of the triangle, and the length of tangents from a point to the incircle can also be calculated if the distance from the point to the incenter is known. Understanding these concepts is crucial for solving problems related to the incircle in trigonometry and geometry.