Question

In an equilateral triangle, prove that the centroid and circumcentre of triangle coincide.

Answer 1

Given: Triangle ABC is equilateral, so AB = BC = CA.

To Prove: The centroid and the circumcentre of triangle ABC coincide.

Proof (Step-wise):

1. Let D be the midpoint of BC.

Then BD = DC by definition of midpoint.

2. Consider triangles ABD and ACD.

We have

AB = AC   ...(Given)

BD = DC   ...(From step 1)

AD = AD   ...(Common side).

Hence, ΔABD ≅ ΔACD by SSS congruence.

3. From the congruence in step 2,

∠ADB = ∠ADC

These two adjacent angles sum to 180°, so each equals 90°.

Therefore, AD ⟂ BC.

4. Since D is the midpoint of BC and AD ⟂ BC, AD is the perpendicular bisector of BC.

But AD is also a median by construction, it joins vertex A to midpoint D.

5. By the same argument, cyclically permuting vertices, the medians from B and C are also perpendicular bisectors of the opposite sides.

6. The centroid is the common intersection point of the three medians; the circumcentre is the common intersection point of the three perpendicular bisectors.

Because each median is also a perpendicular bisector, their intersection is the same point.

Therefore, in an equilateral triangle, the centroid and the circumcentre coincide.