Transformations

Move the picture until the problem is easy.

At the highest level you stop solving the configuration and start changing it. A homothety (scaling about a centre) maps lines to parallel lines and circles to circles: the natural response to tangent circles or parallel lines.

A spiral similarity (rotation plus scaling) explains almost every 'two similar triangles hiding in the figure' situation; its centre is the second intersection of two well-chosen circumcircles.

Inversion is the heavy artillery: $OP \cdot OP' = k^2$ sends circles through the centre to straight lines. Choose the centre where the most circles meet, and a tangle of circles unfolds into lines.

Where it appears on Lemma: Level 8 (advanced configurations).

Train it on Lemma

Transformations unlocks at Level 8 of Lemma's eight-level ladder, with lessons that teach it and drills that make it stick. 125 lessons, 1206 curated problems and unlimited generated practice at six difficulties. Free to start, no card, and every paid plan opens with 3 free days.

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