Inscribing equilateral triangles in convex curvesLet C be a smooth, convex, closed curve, i.e., one without endpoints, and such that a line segment joining any two points on C lies inside C. (An ellipse or an oval are examples. "Smooth" means it has a tangent line at each point.)Let P be any point on C. Show convincingly that you can always find two other points Q and R on C such that PQR is an equilateral triangle. (Try some sketches.)My attempt:Let us denote by D the region inside of C, and let TP denote the set of all equilateral triangles, one of the vertices of which being P. I considered then the set S := { a ⩾ 0 : there exists a triangle Δ ∈ T P with side length a , contained entirely in D ¯ } .I tried showing that S is a closed interval of the form [0,d], and then showing that a maximal triangle of side length d must be touching the curve at three points. However, I couldn't complete the argument.I understand that being an introductory textbook, it is not expected from a reader to provide a rigorous proof here, but I'm curious to see what such a proof would like.

Question

Inscribing equilateral triangles in convex curvesLet C be a smooth, convex, closed curve, i.e., one without endpoints, and such that a line segment joining any two points on C lies inside C. (An ellipse or an oval are examples. &quot;Smooth&quot; means it has a tangent line at each point.)Let P be any point on C. Show convincingly that you can always find two other points Q and R on C such that PQR is an equilateral triangle. (Try some sketches.)My attempt:Let us denote by D the region inside of C, and let TP denote the set of all equilateral triangles, one of the vertices of which being P. I considered then the set   S  :=  {  a  ⩾  0  :      there exists a triangle           Δ      ∈                        T                P               with side length           a        , contained entirely in                   D        ¯              }  .I tried showing that S is a closed interval of the form [0,d], and then showing that a maximal triangle of side length d must be touching the curve at three points. However, I couldn&#039;t complete the argument.I understand that being an introductory textbook, it is not expected from a reader to provide a rigorous proof here, but I&#039;m curious to see what such a proof would like.

pararevisarii · Accepted Answer

Step 1Fix point P. From P let Q traverse the curve in a clockwise direction (starting at P and taking PQ to be the tangent at P) and let PR be a line at       60          ∘       clockwise from PQ, with R the point of intersection with the curve.As Q traverses the curve until PR reaches its limit as the tangent at P the triangles PQR have a sixty degree angle at P. PQ starts at length zero so that   P  Q  −  P  R is negative. PR ends at zero so   P  Q  −  P  R is positive. Given (and to be rigorous you&#039;d have to prove this) that   P  Q  −  P  R varies continuously, it must take a zero value.Step 2Another fact you&#039;d have to prove to be rigorous is that R exists - if you had a corner at P with angle less than sixty degrees, there would be no equilateral triangle (but there is no tangent at a corner, so you have to use the properties of the tangent).

Answered question

Answer & Explanation

New Questions in High school geometry