5
$\begingroup$

diagram

Given a circle, a point $H$ outside the circle, segments $\overline{HE}$ and $\overline{HT}$ tangent to the circle at $E$ and $T$, respectively, and points $I$ and $G$ on the circle such that $I$, $G$, and $H$ are collinear (all as shown above), knowing the measures of $\angle EHG$ and $\angle GHT$ (call them $\alpha$ and $\beta$, respectively, for convenience) determines the measures of each of the four arcs on the circle.

Is it possible to compute the measures of the minor arcs $EI$, $IT$, $TG$, and $GE$ in terms of $\alpha$ and $\beta$ without using trigonometry?

  • 0
    I have a dumb question: "without using trigonometry" - why?2010-12-08
  • 1
    @J.M.: Because I was constructing a problem aimed at the high school geometry level. I wrote it, thought I had enough information, failed to solve the problem using the methods I had expected to work, then verified that the problem was actually determined.2010-12-08
  • 0
    See, I thought I had something, but then you said trig is verboten, so I got curious about this restriction. If so, I got nothin'...2010-12-08
  • 0
    Another thing: I parse from your comment that the system there is that trig comes much later than "high school geometry". Am I correct?2010-12-08
  • 0
    @J.M.: It depends on how one defines "trig"--technically, trig as applied only to right triangles, would be allowed, but still isn't what I wanted in the problem. I'd describe as typical for the U.S. to cover definitions of sine, cosine, and tangent relative only to acute angles in right triangles as standard in Geometry (typically in grade 9 or 10); the definition of secant, cosecant, and cotangent, and the extension to circular functions (functions of rotations rather than of acute angles) is usually in Advanced Algebra or Precalculus (the year following Geometry or the year after that).2010-12-08
  • 0
    (As to being able to solve it with trig, I was pretty sure at the time I was first working through this, that I had a solution that relied on the Law of Cosines or the Law of Sines, but it eludes me now.)2010-12-08
  • 0
    The "far arc / near arc" formulas would seem to be the keys to a trig-free analysis: http://www.mathwarehouse.com/geometry/circle/tangents-secants-arcs-angles.php2010-12-08
  • 0
    @Day Late Don: Those are the formulas that I expected to yield a solution, but I couldn't get enough independent equations to resolve the unknowns (there are essentially 3 unknowns as knowing 3 of the arc measures determines the 4th).2010-12-08
  • 0
    Edited title, added [trigonometry] tag in case aficionados of that subject care to comment.2010-12-09

3 Answers 3