Orthocentre of a triangle is the point where the three angle bisectors of a triangle meet.

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.

Construct $\Delta XYZ$ such that $XY=6.7$$\text{cm}$, $YZ=5.8$$\text{cm}$, $XZ=6.9$$\text{cm}$. Construct its incircle.

Construct $\Delta NTS$ where $\mathrm{NT}=5.7 \mathrm{cm}$, $TS=7.5$cm and $\angle NTS=110°$ and draw incircle and circumcircle of it.