Circumcircle of a Triangle

We will discuss here the Circumcircle of a Triangle and the circumcentre
of a triangle.

A tangent that passes through the three vertices of a
triangle is known as the circumcircle of the triangle.

When the vertices of a triangle lie on a circle, the sides
of the triangle form chords of the circle.

Hence, the centre of the circle is located at the point of intersection of the perpendicular bisectors of the sides of the triangle. This point is known as the circumcentre of the triangle. The radius of the circumcircle is equal to the distance between the circumcentre and any one of the three vertices of the triangle. The circumcentre of a triangle is equidistance from the three vertices. In each of the given figures, the circumcircle of ∆XYZ is the circle with centre O and radius equal to OX, or OY, or OZ.

If ∆XYZ is an acute-angled triangle, as in (i), the circumcentre lies inside the triangle.

If ∆XYZ is a right-angled triangle, as in (ii), the circumcentre
lies on the hypotenuse of the triangle (since, the angle in a semicircle is a
right angle).

If ∆XYZ is an obtuse-angled triangle, as in (ii), the circumcircle lies outside the triangle. ## You might like these

• Here we will solve different types of Problems on relation between tangent and secant. 1. XP is a secant and PT is a tangent to a circle. If PT = 15 cm and XY = 8YP, find XP. Solution: XP = XY + YP = 8YP + YP = 9YP. Let YP = x. Then XP = 9x. Now, XP × YP = PT^2, as the

• We will solve some Problems on two tangents to a circle from an external point. 1. If OX any OY are radii and PX and PY are tangents to the circle, assign a special name to the quadrilateral OXPY and justify your answer. Solution: OX = OY, are radii of a circle are equal.

• The solved examples on the basic properties of tangents will help us to understand how to solve different type problems on properties of triangle. 1. Two concentric circles have their centres at O. OM = 4 cm and ON = 5 cm. XY is a chord of the outer circle and a tangent to

• We will discuss circumcentre and incentre of a triangle. In general, the incentre and the circumcentre of a triangle are two distinct points. Here in the triangle XYZ, the incentre is at P and the circumcentre is at O. A special case: an equilateral triangle, the bisector

• We will discuss here the Incircle of a triangle and the incentre of the triangle. The circle that lies inside a triangle and touches all the three sides of the triangle is known as the incircle of the triangle. If all the three sides of a triangle touch a circle then the