A circle is inscribed in the triangle if the triangle's three sides are all tangents to a circle. In this situation, the circle is called an inscribed circle, and its center is called the inner center, or incenter.
Since the triangle's three sides are all tangents to the inscribed circle, the distances from the circle's center to the three sides are all equal to the circle's radius. Thus, in the diagram above,
where denotes the radius of the inscribed circle.
Also, since triangles and share as a side, and they are in RHS congruence. Therefore Using the same method, we can also deduce and
Another important property of circumscribed triangles is that we can think of the area of as the sum of the areas of triangles and Since the three triangles each have one side of as the base, and as the height, the area of can be expressed as
In conclusion, the three essential properties of a circumscribed triangle are as follows:
- The segments from the incenter to each vertex bisects each angle.
- The distances from the incenter to each side are equal to the inscribed circle's radius.
- The area of the triangle is equal to where is the inscribed circle's radius.
In the above diagram, circle of radius 3 is inscribed in If the perimeter of is 30, what is the area of
The area of a circumscribed triangle is given by the formula
where is the inscribed circle's radius. Therefore the answer is
In the above diagram, circle is inscribed in where the points of contact are and If and what is the perimeter of
Since the circle is inscribed in we have
Therefore, the perimeter of is
In the above diagram, point is the incenter of If and what is
Since is the incenter of , we know that
Since the three angles of a triangle sum up to we have
Thus, the answer is
In the above diagram, circle is inscribed in triangle If what is
Drawing an adjoint segment gives the diagram to the right:
Now we know that
so the answer is
In the above diagram, point is the incenter of The line segment passes through and is parallel to If and what is
Since is the incenter of and is parallel to and are isosceles triangles. Thus, and
Then, we have
Thus, the answer is