Question

The circumcenter is the center of the. circle

Answer 1

Answer: Another name of circumcenter is, " Circumscribed circle ". The circumcenter is the center point of the circumcircle drawn around a polygon. The circumcircle of a polygon is the circle that passes through all of the polygon's vertices & the center of that circle is called the circumcenter. All polygons that have circumcircles are known as cyclic polygons. However, all polygons does not have a circumcircle. Only regular polygons, triangles, rectangles, and right-kites can have the circumcircle and thus the circumcenter as well.

You may find that the circumcenter of a triangle as the most common thing asked in exams and it is generally what schools begin with. So, some brief information about the circumcenter of a triangle is given below. You may safely ignore them if you haven't learned them yet.

What is the Circumcenter of a Triangle?

The circumcenter of triangle can be found out as the intersection of the perpendicular bisectors (the lines that are at right angles to the midpoint of each side) of all sides of the triangle. This means that the perpendicular bisectors of the triangle are concurrent (meeting at one point). All triangles are cyclic and hence, can circumscribe a circle, therefore, every triangle has a circumcenter.

Properties of Circumcenter of Triangle

Property 1: All the vertices of the triangle are equidistant from the circumcenter.

Property 2: All the new triangles formed by joining O to the vertices are Isosceles triangles.

Property 3: Circumcenter lies at the midpoint of the hypotenuse side of a right-angled triangle

Property 4: In an acute-angled triangle, circumcenter lies inside the triangle

Property 5: In an obtuse-angled triangle, it lies outside of the triangle

Note- Location for the circumcenter is different for different types of triangles.

How to construct Circumcenter of a Triangle?

The circumcenter of any triangle can be constructed by drawing the perpendicular bisector of any of the two sides of that triangle. The steps to construct the circumcenter are:

• Step 1: Draw the perpendicular bisector of any two sides of the given triangle.

• Step 2: Using a ruler, extend the perpendicular bisectors until they intersect each other.

• Step 3: Mark the intersecting point as P which will be the circumcenter of the triangle. It should be noted that, even the bisector of the third side will also intersect at P.

Circumcenter Formula

P(X, Y) = [(x1 sin 2A + x2 sin 2B + x3 sin 2C)/ (sin 2A + sin 2B + sin 2C), (y1 sin 2A + y2 sin 2B + y3 sin 2C)/ (sin 2A + sin 2B + sin 2C)]

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Answer 2

The circumcenter is the center of the circle which goes through the triangle's vertices, so the circumcenter of the triangle and the center of that circumscribed circle MUST be the same point.

The same goes for the incenter and the center of the inscribed circle, though these will not, in general, be the same point as the circumcenter.