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Q&A

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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3 comments

Does your kid know about 30-60-90 triangles? Knowing that the short side is half the length of the hypotenuse would be enough. r~~‭ about 1 month ago

@r~~ oh good point. Yes, I think so, actually. msh210‭ about 1 month 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? Peter Taylor‭ 12 days ago

1 answer

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This image matches the description in the question (note that, in violation of what I consider to be conventional, $O$ is not the centre of the circle but the midpoint of $AD$). I add a perpendicular to $AD$ from $O$ which intersects $AC$ at $G$, and lines $OF$ and $DG$ which intersect at $H$.

As described above

The key idea is to show that $AOG$, $DOG$, $DFG$ and $CDF$ are all congruent. Then since the area of a triangle is a half of the base times the height, $$\tfrac{1}{2} \overline{AF} \cdot \overline{DF} = 3 \cdot \tfrac{1}{2} \overline{CF} \cdot \overline{DF}$$

One argument which uses fairly basic lemmata and certainly makes no mention of similar triangles is as follows:

  1. Angle $ACD = 60^\circ$ because it's an angle of an equilateral triangle.
  2. Angle $CAD = 30^\circ$ because AD is a bisector.
  3. $\overline{AO} = \overline{OD}$ by definition of $O$.
  4. Angles $AOG = DOG = 90^\circ$ by definition of $G$.
  5. Angle $AGO = 60^\circ$ by sum of angles in a triangle.
  6. $AOG$ is congruent to $DOG$ by side-angle-side.
  7. Angle $ODG = 30^\circ$ and $OGD = 60^\circ$ in consequence.
  8. Angle $DGF = 60^\circ$ by sum of angles on a line.
  9. $AO$ and $OF$ are radii of the circle, so $AOF$ is isosceles and angle $OFA = OAF = 30^\circ$.
  10. Angle $FOG = 30^\circ$ by sum of angles in a triangle.
  11. Angle $DOF = 60^\circ$ by e.g. sum of angles on a line.
  12. $OD$ and $OF$ are radii of the circle, so $DOF$ is isosceles and angle $ODF = OFD$. But since angles in a triangle sum to $180^\circ$ and $DOF = 60^\circ$ that means that $ODF = OFD = 60^\circ$.
  13. The angles of $OGD$ and $DFG$ are all known and they share edge $DG$, so we have enough to show that they're congruent.
  14. Angle $DFC = 90^\circ$ by sum of angles on a line.
  15. Angle $FDC = 30^\circ$ by e.g. sum of angles in a triangle.
  16. The angles of $DFG$ and $DFC$ are known and they share edge $DF$, so we have enough to show that they're congruent.
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