Circle, external point, tangent rays, and right-angle contact marksLive current inputs
Math & Statistics
Circle Graphing Calculator
Graph a circle and a selected point, classify the point as inside, on, or outside the circle, and construct both tangent contact points when real tangents exist.
Point classification-
Center-to-point distance-
Point power d²-r²-
Tangent length-
First tangent point-
Second tangent point-
Angle between tangents-
Circle equation-
ANALYTIC GEOMETRY GRAPH
Circle, external point, tangent rays, and right-angle contact marks
The coordinate plane redraws the full construction, keeps equal axis scales, and shows why a point inside the circle has no real tangent line.
TANGENCY PROOF
Coordinate checks for both contact points
Each candidate must lie on the circle and its radius must be perpendicular to the tangent segment.
| Object | Coordinates or relation | Circle residual | Dot-product check | Interpretation |
|---|
GRAPH SETUP
Place the point relative to the circle, then read the existence gate
- Enter center coordinates and a positive radius.
- Enter the point from which tangents are considered.
- Use classification before reading tangent values.
- Inspect the equal-axis graph rather than screen aspect ratio.
- Use residual and dot-product rows to verify contact.
WHY INSIDE POINTS FAIL
A tangent contact creates a right triangle with the radius
At tangency, CT is perpendicular to PT, so d²=r²+ℓ². If d<r, the required ℓ² is negative and no real tangent segment exists.
For a point exactly on the circle, both contact points collapse to the point and the tangent is unique rather than a two-ray construction.
EXTERNAL-POINT TANGENTS
Classify by point power before constructing contact points
The squared center distance determines whether tangents exist. For d>r, rotating the center-to-point unit vector by the tangency geometry yields two contact points.
Detailed calculation process and general formulas
d = √[(pₓ-h)²+(pᵧ-k)²]Π = d²-r²ℓ = √ΠT± = C + (r²/d)u ± (rℓ/d)vφ = 2asin(r/d)Symbols, meanings, and units
- C=(h,k)
- circle centercoordinate units
- P=(pₓ,pᵧ)
- selected pointcoordinate units
- d
- distance from center to selected pointlength
- Π
- power of the pointlength²
- ℓ
- length of either tangent segmentlength
- T±
- two tangent contact pointscoordinate units
GRAPH INTERPRETATION
Use classification, length, and contact geometry together
The graph is a construction, not a decorative circle.
Existence
-Point power determines whether there are two, one, or zero real tangents.
Reach
-Both tangent segments from the same external point have equal length.
Contact
-Each displayed contact point satisfies both the circle and perpendicularity equations.
Decision takeaway: Do not extrapolate a real tangent length from an inside point; the power gate must be nonnegative.
Applied decisions
Two uses of point-to-circle tangency
Sight line around a round obstruction
An observer point lies outside a circular keep-out zone.
What the result clarifies: The tangent rays bound the visible or clear corridor.
Belt approach to a pulley
A straight belt segment approaches a circular pulley from an external guide point.
What the result clarifies: Contact coordinates and tangent length define the straight segment geometry.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
The drawing assumes an ideal Euclidean circle and point geometry. Real belts, sight lines, and clearances require thickness, offsets, wrapping direction, and tolerances beyond this two-dimensional tangent construction.
Circle Tangent Graphing Calculator | External Point, Tangent Lines, and Contact Points FAQ
Why are there two tangent points?
An external point supports two symmetric tangent rays around the center-to-point line.
What happens when the point is on the circle?
The two contacts merge and the tangent length becomes zero.
Why is point power useful?
Its sign classifies the point and its square root gives tangent length outside the circle.
Are the graph axes equally scaled?
Yes. Equal scale is required so the circle does not appear as an ellipse.