Inradius, Semiperimeter, and Area - Expii

Given a triangle, what is/are the mathematical relationship/s, if any, between the circle perfectly enclosed by the triangle and the circle perfectly enclosing it? By perfectly I mean perfectly fitting. Could be

In-radius = Semi-perimeter - hypotenuse, Proof

Let r and s be the inradius and semiperimeter of a triangle. Can you please demonstrate (show how) that rs is the area of the triangle? - Quora

What is the relationship between inscribed angles and congruent triangles, and tangents to circles? - Quora

Area of triangle using Inradius (r) and semiperimeter (s)

The Product of the Inradius and Semiperimeter of a Triangle - Wolfram Demonstrations Project

The Product of the Inradius and Semiperimeter of a Triangle - Wolfram Demonstrations Project

Geometry classes, Problem 193. Area of a Triangle, Semiperimeter, Inradius. Math teacher Master Degree. College, SAT Prep. Elearning, Online math tutor, LMS.

Inradius, Semiperimeter, and Area - Expii

What books about solving similar triangles math problems do you recommend so I can completely understand the topic? - Quora

Inradius and Area of Triangle — Find Radius of Inscribed Circle Using Area and Semiperimeter