The construction of incircle is being done by obtaining the point of intersection of two perpendicular of sides and bisector of two angles. 

Construct two circles of radii $4 \mathrm{cm}$ and $2 \mathrm{cm}$ and the distance between their centres is $7 \mathrm{cm}$. Construct a direct common tangent of the circles. (only traces of construction are required).

Consider the following statements:

1. The point of intersection of the perpendicular bisectors of the sides of a triangle may lie outside the triangle.

2. The point of intersection of the perpendiculars drawn from the vertices to the opposite sides of a triangle may lie on two sides.

Which of the above statements is/are correct?

Consider the following statements:

$1.$The orthocentre of a triangle always lies inside the triangle.

$2.$The centroid of a triangle always lies inside the triangle.

$3.$The orthocentre of a right-angled triangle lies on the triangle.

$4.$The centroid of a right-angled triangle lies on the triangle.

Which of the above statements are correct?

Find the in-radius (in cm) of an equilateral triangle whose sides are $6$ cm each.

Two circles are placed in an equilateral triangle as shown in the figure. What ratio of the area of the larger circle to that of the equilateral triangle?