MathLabs

Problem 1

Given a triangle ABC, let I be the incenter. The internal bisectors of angles A, B, C meet the opposite sides in A', B', C' respectively. Prove that 1/4<AI⋅BI⋅CI/(AA′⋅BB′⋅CC′)≤8/271/4<AI\cdot BI\cdot CI/(AA'\cdot BB'\cdot CC')\le8/27.
Step 3 of 4: Parameterize the triangle inequalities
Previous stepNext step
AB=y+z, BC=z+x, CA=x+y,x,y,z>0AB=y+z,\ BC=z+x,\ CA=x+y,\qquad x,y,z>0
Detailed analysis

Set AB+BC−CA=2z, BC+CA−AB=2x, and CA+AB−BC=2y. Then x,y,z are positive, the sides are AB=y+z, BC=z+x, CA=x+y, and p=2(x+y+z).