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Comments on proving relative lengths on a secant

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proving relative lengths on a secant

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My kid was given this question. (Well, my statement of it actually includes some results that my kid had to find in previous parts of the question.)

Triangle $ABC$ is equilateral. $D$ is the middle of side $\overline{BC}$. $AD$ is the diameter of a circle centered at $O$. $\overline{AC}$ meets the circle at $F$. Prove that $AF=3CF$.

[I'd appreciate if someone could add a diagram. I can't at the moment.]

I can think of two ways to do this:

  1. Draw $\overline{DF}$, prove it's an altitude in triangle $ADC$, and use similar triangles to find the sidelengths.
  2. Use the fact that $DC^2=FC\cdot AC$.

Any other methods? I ask because my kid hasn't learned that latter theorem, or similar triangles.

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General comments (3 comments)
General comments
r~~‭ wrote almost 3 years ago

Does your kid know about 30-60-90 triangles? Knowing that the short side is half the length of the hypotenuse would be enough.

msh210‭ wrote almost 3 years ago

@r~~ oh good point. Yes, I think so, actually.

Skipping 1 deleted comment.

Peter Taylor‭ wrote almost 3 years ago

Would I be correct in guessing that the lack of any comment on my answer is because you haven't seen the final version?