Question

Construct the circumcircle and incircle of an equilateral triangle $ABC$ with side $6cm$ and centre 0 . Find the ratio of radii of circumcircle and incircle.

Answer 1

Incentre and circumcentre are same points
Steps for construction:
1. Construct triangle $ABC$ with $AB=BC=CA=6cm$.
2. To draw incircle:
a. Bisect any two angles and let the angle bisectors meet at $I$.
b. From I, draw a perpendicular on $BC$ meeting at point $P$.
c. With I as centre and IP as radius, draw a circle.
3. To draw circumcircle:
a. Draw the perpendicular bisectors of any two sides of the triangle $ABC$.
b. Mark the point of intersection of the perpendicular bisectors as 0
C. Taking $OA$ or $OB$ or $OC$ as radius, draw a circle.
Now, observe that 0 and I are the same points. So, the incentre and circumcentre are same points. Also, take a
ruler and measure the radii of both circles that is IP and OA. You will see that $OA=2IP$.