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.
Train it on Lemma
Phantom Points 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.
Find your levelMore techniques
- Complex Numbers & Coordinates in GeometryWhen the synthetic route stalls, compute.
- TransformationsMove the picture until the problem is easy.
- All 29 techniques →