Construction of the Tangent Line
Identify Key Elements
Given a circle with center O and a point P outside the circle, we aim to construct a tangent from P to the circle.
Connect P to O
Draw the line segment PO, which represents the radial distance from the external point to the circle’s center.
Bisect Segment PO
- Place the compass at P and O with a radius greater than half the length of PO, and draw intersecting arcs above and below PO.
- Draw the perpendicular bisector of PO, marking its midpoint M.
Construct a Semicircle Centered at M
- Set the compass to half the length of PO (that is, the length of MO or MP).
- Draw an arc centered at M with this radius, which will pass through both O and P.
Find Points of Tangency
The intersections of this arc with the given circle determine the points of tangency, Here, T.
Draw the Tangent Lines
Draw line segment PT which is the tangent line.
Animation:
Exactness of the Construction
This method is exact, not an approximation. The key geometric property used is that the angle inscribed in a semicircle is always a right angle (Thales’ Theorem). Since the arc we construct at M forms a semicircle, the segment PT is necessarily perpendicular to OT, ensuring tangency.
Proof of the Construction
Right Angle Condition
By construction, M is the midpoint of PO, and the circle centered at M passes through both P and O.
- Since T₁ and T₂ lie on this arc, the angles ∠PT₁O and ∠PT₂O are both 90 degrees.
- A radius drawn to a tangent is always perpendicular to the tangent at the point of tangency.
- Since OT₁ and OT₂ are radii of the given circle, and we just established that ∠PT₁O and ∠PT₂O are right angles, it follows that PT₁ and PT₂ must be tangent to the circle.
Uniqueness of Tangent Lines
A line passing through an external point and touching a circle at exactly one point is a tangent. Since this construction yields two such lines, they must be the unique tangent lines.