In an equilateral triangle, the circumcenter coincides with which of the following?
The centroid of the triangle
Consider any single vertex of an equilateral triangle. From it, the median (to the opposite midpoint), the altitude (perpendicular to the opposite side), the internal angle bisector, and the perpendicular bisector of the opposite side are all the very same line.
This happens because the triangle has full three-fold symmetry: opposite each vertex the side is symmetric, so all these special lines collapse together.
The circumcentre is the common point of the three perpendicular bisectors, and the centroid is the common point of the three medians.
Since perpendicular bisectors and medians coincide line-for-line, their intersection points must coincide too.
In fact all four classical centres, the circumcentre, incentre, centroid and orthocentre, land on this one point in an equilateral triangle.
So the circumcentre is located exactly at the centroid, at a distance \(\tfrac{2}{3}\) of each median from the vertex.
Hence the circumcenter coincides with the centroid of the triangle.
Similar Questions
Two right-angled triangular blocks, ABC and DEF, have ∠B = ∠E = 90°. If the lengths of the hypotenuses AC and DF are equal, and the sides AB and DE are equal, are the triangles congruent? If so, by what rule?
- A right-angled triangle ABC has legs AB=6 cm and BC=8 cm. An altitude BD is drawn from the vertex B to the hypotenuse AC. What is the length of the altitude BD?
The ratios of the perimeters of two similar triangles are expressed as 3 : 5. Together, their areas total 34 cm². What is the area of the larger triangle?
A right triangle has sides 3, 4, and 5. A smaller triangle is drawn inside it with its vertices on the sides of the larger triangle, such that it is similar to the larger triangle. If its perimeter is 6, what is its area?
A circle is circumscribed about a triangle. If one of the angles of the triangle is 120°, and the radius of the circumscribed circle is r, what is the length of the side opposite the 120° angle?
In triangle ABC, medians AD, BE, and CF intersect at the centroid G. What is the ratio of the area of triangle GAB to the area of triangle ABC?
In △ABC, a line segment DE is parallel to BC, with D on AB and E on AC. If the ratio of the area of △ADE to the area of the trapezoid DECB is 4 : 21, what is the ratio of AD to DB?
- In an urban project, triangular plots with side ratios 3:4:5 are allocated. If the perimeter is 60 m, find the area.
- In $\Delta ABC \sim \Delta XYZ$ and $AB = 4 \text{ cm}, BC = 6 \text{ cm}, XY = 8 \text{ cm}$, what is the length of YZ?
- The Incenter of a triangle is located at $(3, 4)$. The length of the altitude from the Incenter to side AB is $2\text{ units}$. What is the radius of the triangle's incircle?
Important Questions from Triangles, Congruence and Similarity
G is the centroid of the equilateral triangle ABC. If AB = 8√ 3 cm, then the length of AG is equal to:
If sides of a triangle are 12 cm, 15 cm and 21 cm, then what is the inradius (in cm) of the triangle?
What is the area of quadrilateral ABCD?
It is given that ΔABC ~ ΔXYZ and Area ΔABC : Area ΔXYZ = 81 : 25. If AB = 18 cm, BC = 10 cm, CA = 15 cm, then what is the side XZ (in cm)?
Sides of two similar triangles are in the ratio 4 ∶ 9. Area of these triangles are in the ratio:
