Question

Prove that in an equilateral triangle, circumcentre, incentre, centroid, orthocentre will coincide.

Answer 1

Dear Student,

In an equilateral triangle ABC, drop a perpendicular from A on BC to meet at Point D on BC

Now, In $△$ABD and $△$ACD
$\Rightarrow$AB = AC (Sides of equilateral triangle are equal)
$\Rightarrow$AD = AD (Common Side)
$\Rightarrow$$\angle$ADB = $\angle$ADC = 90
Therefore, two triangles are similar by RHS test
Hence, BD = CD
Also, $\angle$BAD = $\angle$CAD

It means that the perpendicular A to BC is also the median of the triangle , also the perpendicular bisector of triangle and also the angle bisector of triangle
Which implies that for an equilateral triangle the median, perpendicular bisector, angles bisector is the same line and hence for the triangle circumcentre, orthocentre, incentre and centroid coincide.

Regards