Ravi Substitution

Trade triangle sides for positive reals and the constraint vanishes.

The idea

Ravi substitution converts an inequality about the side lengths of a triangle into an inequality about unconstrained positive reals.

The side lengths $a, b, c$ of a triangle cannot be chosen freely: they must satisfy the three triangle inequalities $a < b + c$, $b < c + a$ and $c < a + b$.

Proposition (Ravi substitution).

Positive numbers $a, b, c$ are the side lengths of a triangle if and only if there are $x, y, z > 0$ with $a = y + z, \qquad b = z + x, \qquad c = x + y,$ and in that case $x, y, z$ are uniquely determined by $a, b, c$.

The three triangle inequalities on $a, b, c$ therefore collapse into the single condition $x, y, z > 0$, and the standard inequalities for positive reals apply directly.

The new variables have a geometric meaning. From each vertex of a triangle, the two segments drawn to the points where the incircle touches the adjoining sides have equal length, and $x$, $y$, $z$ are those three tangent lengths, as the figure shows.

Ways to work on it

• Walkthrough. Set up a=y+z, b=z+x, c=x+y and see why positivity replaces the triangle inequalities.
• Practice. Read off a Ravi tangent length from a given triangle.
• Hardest. Use the substitution to strip the constraint from a classic triangle inequality.

Not sure where to start? Take the ten-question placement test.