Phantom Points
Define the point you want, then prove it is the one you have.
To show a point has some property, define a phantom by that property, prove satisfies the original construction, then argue uniqueness forces .
It replaces a hard 'prove this configuration' with two easy directions, and it is the standard escape when a synthetic chase stalls because you cannot get hold of the point.
The same trick under a different name — 'assume the conclusion, derive the construction' — appears throughout olympiad geometry write-ups.
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Phantom Points unlocks at Level 8 of Lemma's eight-level ladder, with lessons that teach it and drills that make it stick. 78 lessons, 624 curated problems and unlimited generated practice at six difficulties. Free to start — no card, no trial clock.
Find your levelMore techniques
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