📋 Prerequisites

• Circles Basics
• Congruence & Similarity

Power of a Point is one of the most useful theorems in competition geometry. It relates the lengths of segments created when two lines intersect a circle — whether the intersection is inside the circle (two chords) or outside (two secants, or a secant and a tangent). One formula, three cases.

📚 Key Concepts

What Is the "Power" of a Point?

For a point P and a circle with center O and radius r:

\[ \text{Power of P} = OP^2 - r^2 \]

• If P is inside the circle: power is negative
• If P is on the circle: power is zero
• If P is outside the circle: power is positive (equals the square of the tangent length from P)

The key insight: for any line through P intersecting the circle at X and Y, the product $ PX \cdot PY $ is constant — it equals the power of P (up to sign).

Case 1: Intersecting Chords (Point Inside)

If two chords AB and CD intersect at point P inside the circle:

\[ PA \cdot PB = PC \cdot PD \]

The products of the segments of each chord are equal.

Why: Triangles PAC and PDB are similar by AA (vertical angles + inscribed angles subtending the same arc).

Case 2: Secant-Secant (Point Outside)

If from point P outside the circle, two secants are drawn — one intersecting the circle at A and B, the other at C and D (with PA < PB, PC < PD):

\[ PA \cdot PB = PC \cdot PD \]

\(Whole secant \\times external segment is constant for both secants.\)

Important: PA is the entire length from P to the far intersection B, times the near segment PA. Wait — \(more precisely: external part \\times whole = external part \\times whole.\)

Standard notation: if the secant hits the circle first at A, then continues to B (farther from P):

$ PA_{outside} \times PB_{whole} = PC_{outside} \times PD_{whole} $

Case 3: Tangent-Secant (Point Outside)

If from point P outside the circle, a tangent touches at T and a secant passes through A (near) and B (far):

\[ PT^2 = PA \cdot PB \]

The square of the tangent length equals the product of the entire secant and its external segment.

This is the limit case of the secant-secant theorem where the two intersection points coincide.

Mnemonic: Same Formula, Three Looks

All three cases say the same thing: for a point P and a line through P cutting the circle at X and Y, the product $PX \cdot PY$ is constant (depends only on P and the circle, not the line).

• Inside the circle: two segments, product = constant (negative power)
• \(Outside the circle: external segment \\times whole secant = constant (positive power = tangent^{2})\)
• \(Tangent: both intersection points coincide, so PT \\times PT = PT^{2}\)

Radical Axis (Introduction)

The set of points with equal power with respect to two circles is a line called the radical axis.

• It's always perpendicular to the line connecting the centers
• If the circles intersect, the radical axis is the line through the intersection points
• If the circles are tangent, the radical axis is the common tangent at the point of tangency
• Radical center: For three circles, the three pairwise radical axes meet at one point

✏️ Example Problems