# True? “A rectangle is a subset of a circle if and only if all its vertices are in the circle (or its boundary).”

Is this statement true?

A rectangle is a subset of a circle if and only if all its vertices are in the circle (or in its boundary).

Intuitively it's true, but can someone give me proof?

• If any of the 4 vertices are outside the circle then its obviously not contained within the circle. For all 4 vertices being on or inside the circle use convexity - both a disk and a rectangle are convex shapes. Proving convexity for either shape is trivial. – AlphaNumeric Sep 21 '16 at 14:33
• If you can prove that all the points in a line between two points in a circle are also in the circle that should be sufficient. As AlphaNumeric points out that's the cocept of convexity. I'm not sure I agree that it is trivial as we can't just take "A circle is convex as an act of god" as a given. We need some context. For example are we allowed the analytic definitions of Circle ={(x,y)|(x - a)^2 + (y-b)^2 le r^2} and Rectagle = ... whatever. Then it is easy. Not trivial but easy. – fleablood Sep 21 '16 at 15:50
• I solved it. Yes we are allowed to use the analytic definition of the circle – Marios Gretsas Sep 22 '16 at 13:50