Sprdalo M.

asked • 10/29/19

Help with geometry problem

Let $ABC$ be a triangle. Points $A_1, B_1, C_1$ are, respectively, on sides $BC, AC, AB$ and:


$\frac{AC_1}{C_1B} = \frac{BA_1}{A_1C} = \frac{CB_1}{B_1A} \neq 1$


If $\frac{AB}{A_1B_1} = \frac{BC}{B_1C_1} = \frac{CA}{C_1A_1}$ prove that triangle $ABC$ is equilateral.


I found this problem in the section where problems are based on complex numbers, complex geometry, polynomials,... However, any solution would be welcomed (especially complex one :) ) .Thanks in advance.


In case Latex is not working:


Let ABC be a triangle. Points A1, B1, C1 are, respectively, on sides BC, AC, AB and:

AC1/C1B} = BA1/A1C = CB1/B1A which is not equal to 1

If AB/A1B1 = BC/B1C1} = CA/C1A1 prove that triangle ABC is equilateral.

Mark M.

"AC1/C1B} = BA1/A1C = CB1/B1A != 1" is not a well defined equation. Please revise.
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10/29/19

Sprdalo M.

Sorry, I meant not equal by placing !=
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10/29/19

2 Answers By Expert Tutors

By:

Paul M. answered • 10/30/19

Tutor
5 (19)

BS Mathematics, MD

Sprdalo M.

Can you please explain what sides are in proportion? I don't quite seem to understand why these three triangles are similar...
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10/30/19

Paul M.

tutor
The logic in my answer is not quite right. In fact, I am pretty sure that the argument will be that the ratio for the sides as given can ONLY be true if the triangles are equilateral, but I haven't got that logic written down yet. I will get it to you as soon as I have it.
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10/30/19

Paul M.

tutor
I don't know whether you have an answer yet. I am a bit closer. If the original triangle is NOT equilateral, the only time the inner triangle is similar to the outer triangle is when the sides are split in half. However, if the outer triangle is equilateral, then the result is as stated in the problem, i.e. the inside triangle is also equilateral. I hope that helps.
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11/05/19

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