The Ravi Substitution

Kill the triangle inequality by construction.

If $a,b,c$ are the sides of a triangle, write $a = y+z$ , $b = z+x$ , $c = x+y$ with $x,y,z > 0$ . The triangle inequality becomes automatic — it is now impossible to violate — and the problem turns into an unconstrained inequality in positive reals.

This single substitution converts a large family of 'prove this for triangle sides' problems into standard AM–GM or Schur territory.

Geometrically $x,y,z$ are the tangent lengths from each vertex to the incircle, which is why the substitution feels natural once you have seen the incircle picture.

Where it appears on Lemma: Level 5 (geometry) and Level 7 (inequalities).

Train it on Lemma

The Ravi Substitution unlocks at Level 7 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.

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