Every circle is one stubborn promise.

Pick a centre. Now mark every point that sits exactly the same distance from it. That distance is the radius — and that rule is the whole story.

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Sweep a point around, holding that distance fixed, and the circle simply… happens. Nothing is drawn freehand — it’s the rule made visible.

Centre $(a,\,b)$, radius $r$. Not a new formula — it’s Pythagoras, frozen: the distance from $(x,y)$ to the centre, set equal to $r$ and squared.

Now lay a straight line across it. Solve the two together and you get a quadratic — so a line can cut a circle in two places…

…but slide that line outward and the two crossing points rush toward each other, closing the gap…

…until, at one exact position, they collide into a single point. The discriminant is zero. The line is now a tangent — grazing once.

Zoom into that kiss. The radius to the point of contact meets the tangent at a perfect right angle — always. That one fact cracks nearly every tangent question without a shred of calculus.

That’s the spine of it.

Chords, building circles from points, and twelve graded exam questions with full worked solutions are waiting.

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