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Internal Point

An internal point of a circle is a point located inside the boundary of the circle, meaning its distance from the center is strictly less than the radius. In geometry, it is impossible to draw a tangent to a circle from an internal point because any line passing through it will intersect the circle at two distinct points.

Key Concepts

If a circle has center O and radius r, a point P is an internal point if the distance OP is less than r.

Zero tangents can be drawn to a circle from an internal point.

Any straight line drawn through an internal point will always intersect the circle at exactly two points, forming a secant or a chord.

The collection of all internal points forms the interior region of the circular plane.

Formula & Equation

d < r

d = distance from the center of the circle to the point, r = radius of the circle

Common Misconceptions

Myth: You can draw exactly one tangent from a point inside the circle if you angle it correctly.

Fact: No tangents can be drawn from an internal point. Any line passing through a point inside the circle will always intersect the circle at two points, making it a secant, not a tangent.

Real World Applications

The bullseye on a dartboard represents an internal point relative to the outer circular boundary of the board.

A pepperoni slice placed near the center of a round pizza acts as an internal point within the circular crust.

Frequently Asked Questions

How many tangents can be drawn from an internal point of a circle?

Zero tangents can be drawn from an internal point. Any line passing through it will cut the circle at two distinct points.

How do you mathematically prove a point is an internal point?

Calculate the distance from the center of the circle to the point. If this distance is strictly less than the radius of the circle, it is an internal point.

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