A circle passing through the three vertices of a triangle is inscribed or circumscribed?

In geometry, the circumscribed circle of a triangle is the only circle that passes through all three vertices of the triangle. This property is valid for any triangle that is not flattened, i.e. whose vertices are not aligned. The center of this circle, called the center of the circumscribed circle, is the point of intersection of the bisectors of the sides of the triangle.

The circumscribed circle is defined as the circle passing through all the vertices of a polygon. In the particular case of a triangle, this circle is always unique. The center of the circumscribed circle, often denoted O, is equidistant from the triangle’s three vertices. This equidistance is due to the fact that O lies on the bisectors of the triangle’s sides, each bisector being the locus of points equidistant from the ends of the segment it intersects.

To draw the circumscribed circle of a given triangle, follow the steps below:

1. Draw the bisectors of the triangle’s sides: for each side, determine its midpoint, then draw the perpendicular line passing through this midpoint. These lines are called bisectors;
2. Identify the point of intersection of the perpendicular bisectors: The three perpendicular bisectors intersect at a single point, which is the center of the circumscribed circle;
3. Draw the circle: Using a compass, place the dry point on the center you’ve found, then open the compass on one of the triangle’s vertices. Then draw the circle through all three vertices.

This method ensures that the circle drawn is the circumscribed circle of the triangle.

It’s important to distinguish between circumscribed and inscribed circles. The inscribed circle of a triangle is the circle tangent to the triangle’s three sides, located inside the triangle. Its center is the point of intersection of the bisectors of the triangle’s angles. Thus, the inscribed circle is contained within the triangle, while the circumscribed circle encompasses the triangle and passes through its vertices.

• Right-angled triangle: In a right-angled triangle, the center of the circumscribed circle is the midpoint of the hypotenuse. In effect, the hypotenuse becomes the diameter of the circumscribed circle, and the center of the circle is therefore located halfway between the two ends of the hypotenuse;
• Equilateral triangle: In an equilateral triangle, the center of the circumscribed circle coincides with the center of the inscribed circle, the center of gravity and the orthocenter, since all these notions overlap at a single point due to the triangle’s perfect symmetry.

The concept of a circumscribed circle is fundamental to geometry and has applications in many fields, including trigonometry, analytical geometry and even engineering.

A circle passing through the three vertices of a triangle is called a circumscribed circle. It is unique to each triangle and plays a crucial role in the study of the geometric properties of triangles. Understanding this concept is essential for a better understanding of plane geometry and its practical applications.

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