# Inscribed equilateral triangle in epitrochoid

Recently I came across a question to show that an equilateral triangle can be inscribed in an epitrochoid (Calculus by Stewart, Chapter 10 Challenge Problems), along with a solution in this link:

http://www.stewartcalculus.com/data/CALCULUS%20Early%20Transcendentals/upfiles/WebChallengeProblems_5ET.pdf

I've tried to do the same question and what I managed to come up with, via an argument on equidistances between points of angular separation $\frac{2\pi}{3}$ using polar coordinates, that every point of the epitrochoid is a vertex of an equilateral triangle with centroid lying on the circle centred at the origin with radius $b$. It's incredibly tedious, but it worked.
What I can't prove, however, is that the triangle must be inscribed within the epitrochoid, like the wording of the solution suggests. The height of the triangle is $4.5r$, where $r$ is as given in the question. The $y$-intercept distance, however, is $6r-2b$. It seems to me that since $0<b<r$, for $b > 0.75r$ the triangle's edges will intersect the epitrochoid as the triangle's vertices sweep out the boundary of the epitrochoid, so not every equilateral triangle must necessarily be inscribed in the epitrochoid.