Tangent and a secant

Parametric equations

The unit circle. The point $P=(p,0)$ lies outside it. One secant leaves $P$ toward the upper left. $A$ is the nearer intersection and $B$ the farther one, so $|PA|$ is the external part and $|PB|$ is the whole secant. The angle is at least $8^{\circ }$ and at most $\arcsin (0.92/p)$. The factor $0.92$ is the same margin as on the two-secants page: the intersections separate before they would meet at a point of tangency.

The upper point of tangency is $T=\left(1/p,\sqrt{1-1/p^2}\right)$. An equally long tangent lies on the lower side. It is not drawn.

$|PT{|}^2$ and $|PA|\cdot |PB|$ are both computed from unrounded lengths measured at the coordinates. The tangent-secant theorem, a case of the power of a point, says they are equal. Values are shown to three decimal places, rounded half up.

Interaction

• Position $p$: move the exterior point; the tangent segment and the two secant distances change together
• Angle: move the angle of the secant; the point of tangency stays where it is

What to notice

• The square of the tangent segment equals this secant’s external part times the whole secant.
• When the secant changes direction, the point of tangency does not move, and the two sides stay equal.
• The tangent segment itself does not change with the secant.

Related

• Two secants
• Secants and tangents

Further reading

• Power of a point (Wikipedia)