Geometry - Constructions (Tangents to a circle, Circumscribing and Inscribing a circle)

Tangent to a Circle: A tangent is a straight line that touches the circumference of a circle at exactly one point, known as the point of contact. A key visual property is that the radius of the circle is always perpendicular to the tangent at the point of contact, forming a $90^{\circ}$ angle.

Tangents from an External Point: From any point outside a circle, exactly two tangents can be drawn to the circle. These two tangents are equal in length ($PA = PB$ where $P$ is the external point). Visually, the line joining the center of the circle to the external point bisects the angle between the two tangents.

Circumscribed Circle (Circumcircle): This is a circle that passes through all three vertices of a triangle. The center of this circle, called the circumcenter, is found by constructing the perpendicular bisectors of at least two sides of the triangle. The distance from the circumcenter to any vertex is the circumradius ($R$).

Inscribed Circle (Incircle): This is a circle that lies inside a triangle and touches all three of its sides. The center of this circle, called the incenter, is found by constructing the angle bisectors of at least two internal angles of the triangle. The perpendicular distance from the incenter to any side is the inradius ($r$).

Construction of Tangent at a Point on the Circle: To construct a tangent at a point $P$ on the circumference, draw the radius $OP$. Then, using a compass, construct a line perpendicular to $OP$ at point $P$. This line is the required tangent.

Construction of Tangents from an External Point: To draw tangents from external point $P$ to a circle with center $O$, first join $OP$. Construct the perpendicular bisector of $OP$ to find its midpoint $M$. Draw a second circle with center $M$ and radius $MO$. The points where this second circle intersects the original circle are the points of contact for the tangents from $P$.

Regular Hexagon in a Circle: To inscribe a regular hexagon in a circle, the radius of the circle is used as the side length. By placing the compass pointer on the circumference and marking off arcs equal to the radius, exactly six points are created. Connecting these points forms a hexagon where each internal angle is $120^{\circ}$.