In an obtuse triangle $ABC$ which one of the following pairs of point's with respect to the triangle $ABC$ must lie outside the triangle?

A semi - circle of diameter $1$ unit sits at the top of a semi-circle of diameter $2$ units. The shaded region inside the smaller semi-circle but outside the larger semi-circle is called a lune . The area of the lune is -

In the figure, $ABCD$ is a unit square. A circle is drawn with centre $O$ on the extended line $CD$ and passing through $A.$ If the diagonal $AC$ is tangent to the circle, then the area of the shaded region is

Let $A_1, A_2$ and $A_3$ be the regions on $R^2$ defined by

$A_1=\left\{\left(x,y\right):x\ge 0,y\ge 0,2x+2y-x^2-y^2>1>x+y\right\}$

$A_2=\left\{\left(x,y\right):x\ge 0,y\ge 0,x+y>1>x^2+y^2\right\}$

$A_3=\left\{\left(x,y\right):x\ge 0,y\ge 0,x+y>1>x^3+y^3\right\}$

Denote by $\left|A_1\right|, \left|A_2\right|$ and $\left|A_3\right|$ the areas of the regions $A_1, A_2$ and $A_3$ respectively. Then

Let $AB$ be a sector of a circle with centre $O$ and radius $d.\angle AOB=\theta \left(<\frac{\pi }{2}\right)$ and $D$ be a point on $OA$ such that $BD$ is perpendicular$OA$. Let $E$ be the mid-point of $BD$ and $F$ be a point on the arc $AB$ such that $EF$ is parallel to $OA$. Then, the ratio of length of the arc$AF$ to the length of the arc $AB$ is

Prove that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent.

In the adjoining figure,$\mathrm{OPQR}$ is a square.$\mathrm{A}$ circle drawn with center $\mathrm{O}$ cuts the square in $\mathrm{X}$ and $\mathrm{Y}$ Prove that $\mathrm{QX}=\mathrm{QY}$

In the given figure, $△ABC$ is an isosceles triangle in which  $AB=AC$ and a circle passing through $B$ and $C$ intersects $AB$ and $AC$ at $D$ and $E$ respectively.

Prove that $DE\parallel BC$.

$ABCD$ is a rectangle. Prove that the center of the circle through $A, B, C, D$ is the point of intersection of its diagonals.

In the given figure, $\mathrm{AB}$ is a diameter of a circle with center $\mathrm{O}$ and $\mathrm{DO}\parallel \mathrm{CB}$. If $\angle \mathrm{BCD}=120°$ calculate $\angle \mathrm{ADC}$

In a huge park, people are concentrated at three points as shown in the figure.
A: Where there are different slides and swings for children.
B: near which a man made lake is situated.
C: Which is near to a large parking and exit.
Where should an ice-cream parlor be set up so that maximum number of persons can approach it?

A line intersecting the circle in two points is called