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I am having trouble identifying the height of each triangle.

dxiv
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Lily
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1 Answers1

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Hint: triangles with bases on the same line and a common third vertex have areas proportional to their bases (because they share the same height). Use that to prove that:

$$\frac{AC'}{C'B} = \frac{area(\triangle CAC')}{area(\triangle CC'B)} = \frac{area(\triangle PAC')}{area(\triangle PC'B)}$$

Then note that $area(\triangle CPA) = area(\triangle CAC') - area(\triangle PAC')$, and similar for $\triangle CPB$.

Finally, remember that for equal fractions (with $b \ne d)$:

$$\lambda = \frac{a}{b} = \frac{c}{d} \quad \implies \quad \lambda = \frac{a-c}{b-d}$$

So:

$$\frac{AC'}{C'B} = \frac{area(\triangle CAC') - area(\triangle PAC')} {area(\triangle CC'B) - area(\triangle PC'B)} = \frac{area(\triangle CPA)}{area(\triangle CPB)}$$

which concludes the proof.

dxiv
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  • I am having trouble finding the area(triangle CPA) because I do not know the height. – Lily Oct 03 '16 at 23:13
  • You don't need the heights of $\triangle CPA$, see the new edit. – dxiv Oct 03 '16 at 23:15
  • So you are saying that triangle CPA and CPB are similar? – Lily Oct 03 '16 at 23:28
  • No, I meant that a similar equality holds for $\triangle CPB$ namely that $area(\triangle CPB) = area(\triangle CC'B) - area(\triangle PC'B)$. – dxiv Oct 03 '16 at 23:36
  • I am still having trouble with this. I know i need to prove some two similar triangles to find the area of triangle CPA and CPB. I still do not know how you got the ratios. – Lily Oct 03 '16 at 23:43
  • My professor has indicated to using area formulas to find the area of triangle CPA and CPB. – Lily Oct 03 '16 at 23:43
  • And then use similar triangle properties to get the conclusion that those areas are equal to AC'/C'B. – Lily Oct 03 '16 at 23:44
  • Could we look at those two right triangles? I think I understand better from there. – Lily Oct 03 '16 at 23:45
  • I know those two right triangles are similar by AA – Lily Oct 03 '16 at 23:45
  • My answer gives a different solution than what your professor suggested. Next time, if you are looking for some particular solution, only, then better indicate that in the body of the question. – dxiv Oct 03 '16 at 23:53
  • Ok. Thank you for trying to help. I will just ask my professor then. – Lily Oct 03 '16 at 23:56