Geometry: Basic to Advanced

The subject with no procedure

Geometry defeats students who are strong elsewhere, and the reason is structural. In algebra, a student who knows the method can execute it. In geometry, a student can know every theorem in the syllabus and still have no idea which one applies to the figure in front of them.

That gap does not close by learning more theorems. It closes with volume and with a systematic habit, and this course is built around installing that habit.

The diagram is a working tool

Students consistently underrate this. A diagram drawn carelessly hides the answer; a diagram drawn accurately frequently gives it away.

If three points look collinear in an accurate figure, they almost certainly are — and now you know what to prove rather than wondering what is true. If a point appears to lie on a circle, test it. We insist on accurate construction from the first week, with compass and ruler or properly with software, and students are regularly startled by how much of the difficulty was self-inflicted.

Systematic angle chasing

The core skill the subject rewards is not cleverness but method: draw accurately, mark every known angle and length, look for the standard configurations, and ask what the conclusion would require to be true.

Students who hope rather than search can solve easy problems and stall completely on medium ones. We work angle chasing as a systematic procedure from the basic stage, because it is the foundation everything else sits on.

Circle theorems carry the subject

If a student has time for one unit, it is this one. Cyclic quadrilaterals and the test for concyclicity appear in a large share of competition geometry, and the alternate segment theorem and tangent properties are close behind.

Immediately after comes power of a point and the radical axis, which is the technique that collapses a whole class of problems. Configurations that look hopeless synthetically resolve in a few lines once a student thinks to compute a power or locate a radical centre. These two units together make most olympiad geometry attemptable.

Choosing your method

By the advanced stage a student should be choosing deliberately between synthetic, coordinate, trigonometric and vector approaches.

Coordinate geometry will grind out many problems that synthetic methods solve in three lines — and under time pressure, grinding is sometimes exactly right. But some configurations become appalling in coordinates and nearly trivial under homothety or inversion. Students left with one tool use it everywhere; we teach the judgement of which to reach for, which is a skill in itself.

Inversion, last and worth it

Inversion in a circle is a genuine change of perspective rather than another formula. Circles become lines, tangency is preserved, and a configuration crowded with circles can become one with two lines and an obvious conclusion.

It is placed last because it requires the circle work to be secure, and it is the point at which strong students stop finding olympiad geometry intimidating. Cross-ratio, harmonic conjugates and poles and polars follow as an introduction to projective thinking.

What to do when you are stuck

In geometry being stuck is the normal state, not a sign of failure, and students who have not been told this conclude they are bad at the subject.

So we teach an explicit checklist for the moment nothing is obvious. Redraw the figure larger and more accurately. Mark every angle and length you actually know, not the ones you assume. Ask what the conclusion would require to be true, and work backwards from it. Look for a cyclic quadrilateral, because one is usually hiding. Try the point you have not used, since competition problems rarely include a redundant condition. Consider whether the configuration would simplify under a transformation.

None of this is clever. It is a procedure, and having one is the difference between a student who stares at a hard problem for twenty minutes and a student who makes progress on it.

Who this suits

Students from Class 9 upward through undergraduate study, with school geometry and basic algebra. The course rebuilds the angle and triangle work before extending it, so a student who has always found geometry opaque can start here.

Classes are live and online in groups of at most eight. Students construct and work problems during the session, because watching someone else see the configuration teaches very little.