Angle chasing: when to stop and what to reach for instead

Angle chasing is the right opening move in almost every olympiad geometry problem, and that is exactly why it is so easy to overrun. The chase rarely fails outright. It finishes — quietly, several lines before you notice — and then you keep labelling, producing angles you already knew in a slightly different order, while the clock runs.

So the question is almost never whether angle chasing was the wrong idea. It usually was not. The question is how to tell that it has already done everything it can do, and what the problem is asking you to try instead.

What a chase can actually produce

Angle chasing propagates one kind of fact: this angle equals that angle, or these two angles sum to $180^\circ$. Everything the technique knows follows from the inscribed angle theorem and the parallel-line transversal, and both of those are statements about angles alone.

Two consequences follow, and between them they explain every stall.

It concludes angle facts, concyclicity, and nothing else. Concyclicity looks like an exception but is not: four points are concyclic exactly when two of them subtend equal angles at the other two, so a concyclicity proof is an angle proof wearing a circle. Opposite angles of a cyclic quadrilateral summing to $180^\circ$ is the whole technique in one problem, and the angle in a semicircle is the same engine running on a diameter.

It is blind to size. Scale a figure by any factor and every angle in it is unchanged, so no chain of angle equalities can pin down an actual length: the chase cannot tell your triangle apart from one seven times larger. Ratios of lengths do survive scaling, which is why those stay reachable — but only through a conversion step, never from angles by themselves.

Four signals to stop

1. The conclusion is not an angle. If what you must prove names a ratio, a product, a concurrency, a length or an inequality, the chase cannot be your solution. It can still be the first half of one, which is the next section.
2. A full circuit added nothing new. Go round the diagram once, naming every angle you can. If a second circuit yields no equality you did not already have, the configuration is saturated and every further line is a restatement of one above it.
3. No circle and no parallel line. With neither, there is nothing to propagate along, and angle sums inside triangles are bookkeeping rather than progress.
4. You cannot name one angle at the point you need. A point defined as the crossing of two lines that nothing else meets is invisible to the chase: no arc runs through it, no transversal cuts it.

Signal 2 is the one people miss, and missing it is expensive, because a saturated diagram still feels productive. You are writing, the page is filling, and every line is true.

What the conclusion says to reach for

This is the part that turns stop into a decision. Trying something else almost always means converting, and the equal angles you already found are the raw material the conversion needs. Read the shape of the conclusion and take the row that matches it.

• A ratio of lengths → similarity. The equal angles you just proved are the AA you need. The angle bisector theorem, where the bisector from $A$ cuts $BC$ in the ratio $AB:AC$, is the standard example: angle facts in, a length ratio out.
• A product of lengths → power of a point. Two secants through $P$ cut off equal inscribed angles, those give similar triangles, and the similarity gives $PA \cdot PB = PC \cdot PD$. The useful part is that this product is indifferent to which line through $P$ you drew: it is $|OP^2 - r^2|$ for all of them at once.
• A concurrency or a collinearity → Ceva and Menelaus. Turn your angles into ratios first, then multiply. Three cevians meet at a point exactly when the product of the three side ratios is $1$, and a triple that misses concurrency misses $1$.
• An inequality or a maximum → leave the chase entirely. Angles carry no order information about lengths, so no amount of labelling will ever produce a $\le$. Ptolemy's inequality is the model here: it comes out of an inversion, and it is tight precisely when the four points are concyclic in order.
• A named centre, or a ratio along a line → coordinates, vectors or complex numbers. The Euler line, with orthocentre, centroid and circumcentre collinear and $HG = 2\,GO$, is a few lines of vector algebra and essentially unreachable by chasing angles.
• A point you cannot get hold of → phantom points. Define the point by the property you want it to have, then prove it is the one the construction already gave you.

The stopping rule in one line

Before you write another angle, ask whether it is either new or the AA you are about to use. If it is neither, the chase is finished, and the conclusion has been telling you which tool to pick up since the moment you read it.

On a timed paper, give the chase an explicit budget — a third of the time you have allotted to the problem is a reasonable default — and switch when it expires. Coordinates are ugly and finite. A stalled chase is elegant and unbounded, and only one of those two fits inside a contest.

How to train the recognition

Classify, do not solve. Take twenty geometry problems and write a single line on each: what kind of object the conclusion is, and therefore what has to finish it. You will get through all twenty in the time one full write-up takes, and classification is the part you are actually short of.

Lemma teaches angle chasing as a named technique with its exit routes attached, rather than as a habit you pick up and never put down. The problem archive has the statements above in full, with difficulty and expected time on each, and the daily problem posts a new one every day in three tiers, no account needed.