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Toolkit 68

Circles, Tangents, and Radical Axis

1. Tangent and Radius

Radius perpendicular to tangent line at point of tangency

The radius is perpendicular to the tangent line.

OP⊥tangentOP\perp\text{tangent}

2. Two Tangents from an External Point

Two tangents from external point A meeting circle at B and C

If two tangents are drawn from the same external point, their lengths are equal.

AB=ACAB=AC

3. Radical Axis

Two intersecting circles with radical axis through intersection points

When two circles intersect, draw the radical axis. It is usually useful in angle chasing.

The radical axis is the line passing through the intersection points of the two circles.

4. Two Tangent Circles

When two circles are tangent, the centers and the tangent point are collinear.

Two externally tangent circles with centers O1, O2 and tangent point P

For externally tangent circles:

O1, P, O2 are collinearO_1,\,P,\,O_2\ \text{are collinear}
O1O2=R1+R2O_1O_2=R_1+R_2

5. One Circle Inside Another (Internally Tangent)

Small circle inside larger circle, internally tangent at point P

For internally tangent circles:

O1, O2, P are collinearO_1,\,O_2,\,P\ \text{are collinear}
O1O2=R2−R1O_1O_2=R_2-R_1
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