A unified theorem for all chord, secant, and tangent relationships from a single external point.
The Mnemonic
One Big Idea, Three Specific Cases
"Power of a Point" is the umbrella name for a family of related theorems describing what happens when lines through a single point (inside OR outside a circle) intersect that circle. The Chord Product Theorem was actually the FIRST case of this bigger idea — it covers a point INSIDE the circle. Two more cases cover a point OUTSIDE the circle.
Two secants from an external point: (whole secant 1) × (outside part of secant 1) = (whole secant 2) × (outside part of secant 2). A secant and a tangent from an external point: (whole secant) × (outside part of secant) = (tangent length)².
Two secants from external point P — (whole)×(outside part) is equal for both secants
💡 Memory Trick
"A unified theorem for all chord, secant, and tangent relationships." All three cases follow the same underlying pattern: multiply the WHOLE length of a line from the external point by its OUTSIDE (external) piece — and that product is the same no matter which secant or tangent from that same point you use. The tangent case is just the special situation where the 'whole' and 'outside part' happen to be the exact same segment, since a tangent only touches the circle once.
Why It Works
Why All Three Cases Are Really the Same Theorem
Each version of Power of a Point can be proven the same way the Chord Product Theorem was: draw the two "cross-connecting" segments between the far endpoints, form two similar triangles (again using shared/vertical angles and the Inscribed Angle Theorem), and the resulting proportion always simplifies to a whole-times-outside-part relationship.
The tangent case is a beautiful limiting case of the secant case: imagine a secant line slowly rotating until its two intersection points with the circle merge into a single point — at that exact moment, it becomes a tangent line, and the secant's "whole × outside" pattern collapses into (tangent length) × (tangent length) = (tangent length)², since the "outside part" and the "whole" length are now identical.
Using It In A Proof
Identifying Which Case Applies
Correctly identifying which of the three Power of a Point cases fits a given diagram is the essential first step, since each case has a slightly different equation shape.
1
Locate the point
Inside the circle → use the Chord Product Theorem (a·b = c·d). Outside the circle → move to step 2.
2
Identify what lines pass through the external point
Two secants → use whole×outside = whole×outside. One secant and one tangent → use whole×outside = tangent².
3
Set up the equation using the correct segments
Carefully identify the "whole" length (external point to the far intersection) and the "outside part" (external point to the near intersection) for each line.
Full Worked Example
Solving a Secant-Tangent Problem
Given: From external point P, a tangent segment PT measures 12. A secant from P passes through the circle, with the near intersection point at distance 8 from P. Find: the whole length of the secant (from P to the far intersection point).
1
Identify the case
One tangent (PT=12) and one secant (near part = 8, whole part = unknown, call it x) from the same external point — this is the secant-tangent case.
2
Set up the equation
(whole secant) × (outside part) = (tangent)², so x × 8 = 12².
3
Simplify
8x = 144.
4
Solve for x
x = 18 — the whole secant length from P to the far intersection point is 18.
Notice the far intersection point is 18−8=10 units past the near intersection point — this inner segment length is sometimes what's actually asked for, so always re-check exactly which length the question wants.
🎯 Quick Worked Example
Two secants from external point P: one has a whole length of 20 and outside part of 5. The other has an outside part of 4. Find its whole length.
1
Set up the equation. 20 × 5 = (whole 2) × 4.
2
Simplify. 100 = 4 × (whole 2).
3
Solve. whole 2 = 25.
📌 Exam Application
The three Power of a Point cases (chords, two secants, secant-tangent) are frequently tested together in the same problem set specifically to check whether students can correctly identify WHICH case a given diagram represents — always start by locating the point (inside or outside) and identifying which types of lines pass through it before setting up any equation.
⚠️ Most Common Power of a Point Mistakes
Trap 1 — Using the wrong case's equation: Two secants use whole×outside = whole×outside; a secant and tangent use whole×outside = tangent² — applying the two-secant equation to a secant-tangent situation (or vice versa) gives a wrong setup entirely.
Trap 2 — Confusing 'whole' length with 'outside' (or 'inside') part: The 'whole' is the FULL distance from the external point to the FAR intersection; the 'outside part' is only to the NEAR intersection — mixing these up, especially when a problem asks for the inner segment specifically, is a common source of error.
✓ Quick Self-Test
1) What are the three cases covered by Power of a Point? 2) How does the Chord Product Theorem relate to Power of a Point? 3) State the equation for the secant-tangent case. 4) A tangent segment measures 9, and a secant's outside part is 3 — find the secant's whole length. 5) Why is the tangent case considered a 'limiting case' of the secant case?