Power of a point: when to use it, and when not to

The statement of power of a point takes one line. Fix a point $P$ and a circle. Every line through $P$ that cuts the circle at two points $A$ and $B$ gives the same product $PA \cdot PB$, no matter which line you picked. Nobody reads that sentence and fails to understand it.

People fail to use it, because no problem announces itself. The paper says $PA \cdot PB$, or it draws a tangent and asks for a length, or it hands you two circles and asks you to prove three lines meet. Nothing in any of that contains the words "power of a point". The whole difficulty sits upstream of the theorem: it is knowing that this is where the theorem goes.

So this is a recognition guide, not an explanation. The technique page has the statement.

One number, and the thing it will not tell you

For a circle with centre $O$ and radius $r$, the power of $P$ is $OP^2 - r^2$. One number, computed before you have drawn a single line, and it controls every line through $P$ at once: the product $PA \cdot PB$ is its absolute value, always.

The sign is the part worth memorising, because it is a recognition signal in its own right. The power is positive strictly outside the circle, negative strictly inside, and exactly zero on it. Which means the technique tells you nothing at all about a point that lies on the circle. If the interesting point in your configuration is on the circle, you are holding the wrong tool and should go back to angle chasing.

The five tells

• A product of two lengths measured along the same line. $PA \cdot PB$, $XM \cdot XN$, anything of that shape. This is the single most reliable signal there is, and it fires whether or not a circle has been mentioned.
• Two products set equal to each other. The moment a problem asks you to prove $PA \cdot PB = PC \cdot PD$, check whether the four points could be concyclic. If they are, the statement is not a fact to be proved so much as a fact to be quoted.
• A tangent, and a length on it. Tangency turns the two secant points into one, and the product collapses to a square: $PT^2 = PA \cdot PB$. A squared length sitting next to a product of two lengths is power of a point in almost every case.
• Two chords crossing inside the circle. Same statement, seen from inside. The crossing point is $P$, its power is negative, and the two products of the pieces are equal.
• Two circles, and a question about collinearity or concurrency. This is the radical axis, and it is the tell people take longest to learn. More on it below.

Reading a configuration

Suppose $P$ sits outside a circle. One secant through $P$ meets it at $A$ and $B$ with $PA = 4$ and $PB = 9$. A second secant meets it at $C$ and $D$ with $PC = 3$. A tangent from $P$ touches at $T$.

The power of $P$ is $4 \cdot 9 = 36$, and it is now fixed for the whole picture. So $PD = 36 / 3 = 12$, and $PT = \sqrt{36} = 6$. Neither the radius nor the position of the centre was needed, and that economy is the point of the technique: one product bought you every other product in the figure.

Notice what happens if you try to move $D$. The four points $A$, $B$, $C$, $D$ are concyclic precisely because the products agree; slide $D$ along its own line by any amount and the product $PC \cdot PD$ strictly changes, so it can no longer equal $36$. The equality of the two products does not merely follow from concyclicity, it pins it down. That is what makes the technique usable in reverse, to prove four points lie on a circle, and it is the shortest route to a concyclicity when the angle route is blocked.

When the circle is not drawn

The hardest instances are the ones with no circle in the diagram. The move is to build one, and there is a standard trigger: a right angle. Since an angle inscribed in a semicircle is right, the converse gives you a circle for free. Any right angle at $A$ subtending a segment $BC$ puts $A$ on the circle with diameter $BC$.

Take a right angle at $A$ in triangle $ABC$, with $H$ the foot of the altitude from $A$. Nobody drew a circle. But $A$ lies on the circle with diameter $BC$, $H$ lies inside it, and the power of $H$ is $-AH^2$ on the altitude and $-BH \cdot HC$ on the side, so $AH^2 = BH \cdot HC$. The geometric mean relation people memorise separately is just power of a point applied to a circle that was never drawn.

Generalise the habit rather than the result: when a configuration hands you a right angle, a tangency, or a pair of equal angles, ask which circle it is secretly announcing. Then ask what point in the problem has an interesting power with respect to it.

Two circles, and the collinearity you get for nothing

Give a point a power with respect to two different circles and subtract. The squared terms cancel, and what is left is linear in the coordinates of the point, so the set of points with equal power to both circles is a straight line: the radical axis. Every point on it has the same tangent length to both circles, and when the circles actually meet, the line is exactly the line through their two intersection points.

That is why a problem with two or three circles and a request for collinearity or concurrency is so often a power problem with no computation in it at all. You are not chasing lengths; you are observing that three radical axes have nowhere else to be.

When it is the wrong tool

Recognition is half negative. Three cases where the products are a distraction:

1. The products do not share a point. $AC \cdot BD$ against $AB \cdot CD$ in a cyclic quadrilateral is Ptolemy, not power of a point. Factors measured from different vertices are a different theorem.
2. The unknown is an angle. Power of a point destroys angle information by design, and it will not give any back.
3. The key point is on the circle. Its power is zero, and zero times anything is a statement about nothing.

Training it

Take twenty circle problems and do not solve any of them. For each one write three lines: which point is $P$, which circle, and what its power buys. You will get through twenty in the time one full solution takes, and pattern recognition is the thing you are actually short of.

Lemma teaches this as a named technique with the three faces kept together, so secants, tangent-and-secant and crossing chords never become three separate memorised facts. The technique index lists it alongside the geometry tools it competes with, the problem archive carries the statements above in full with difficulty and expected time, and the daily problem posts a fresh one every day in three tiers with no account needed.