Construct the orthocentre of the triangle $ABC$, where $AB=5 \mathrm{cm}, BC=4 \mathrm{cm}$ and $AC=5.5 \mathrm{cm}$.

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?

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?

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?

Construct $\Delta ABC$ such that $\angle B=100°, BC=6.4, \angle C=50°$ and construct its incircle.