Incentre and circle formed by it

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In summary, an incentre is the point of concurrency of the angle bisectors of a triangle, formed by drawing the angle bisectors of each angle and finding the point where they intersect. The circle formed by the incentre, called the inscribed circle or incircle, is tangent to all three sides of the triangle and has a radius equal to the distance from the incentre to any side of the triangle. The coordinates of the incentre can be found using a formula involving the coordinates of the triangle's vertices and the lengths of its sides. The inradius, which is the radius of the inscribed circle, can also be calculated using a formula involving the triangle's area and semi-perimeter. The incentre is always located inside the
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Why circle formed by incentre can not lie outside the triangle.
I think because perpendicular drawn from incentre to each side will always be tangent to the circle.
What's your opinion.
 
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What is an incentre and how is it formed?

An incentre is the point of concurrency of the angle bisectors of a triangle. It is formed by drawing the angle bisectors of each angle of the triangle and finding the point where they intersect.

How is the circle formed by the incentre related to the triangle?

The circle formed by the incentre is called the inscribed circle or incircle, and it is tangent to all three sides of the triangle. This means that the radius of the circle is equal to the distance from the incentre to any side of the triangle.

What is the formula for finding the coordinates of the incentre?

The coordinates of the incentre can be found using the formula (ax1 + bx2 + cx3) / (a + b + c) , (ay1 + by2 + cy3) / (a + b + c), where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the three vertices of the triangle, and a, b, and c are the lengths of the sides opposite to those vertices.

What is the relationship between the incentre and the inradius?

The inradius is the radius of the inscribed circle formed by the incentre. It is equal to the distance from the incentre to any side of the triangle. Additionally, the inradius can be calculated using the formula A / s, where A is the area of the triangle and s is the semi-perimeter (half of the perimeter) of the triangle.

Can the incentre lie outside of the triangle?

No, the incentre is always located inside the triangle. This is because the angle bisectors of a triangle must intersect inside the triangle in order to form the incentre. If the incentre was located outside of the triangle, the angle bisectors would not intersect.

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