I was going through some Olympiad math questions, when I came across this question

In $\triangle ABC, AB = AC, \angle A = 120°.$ Points $D, E, F$ are on segments $BC, CA, AB$ respectively such that $CD = CE, BD = BF, DE = 10, FD = 6$. Find area of $\triangle DEF$.

This question left me in a haze, for the fact that the diagram can't be drawn using scale-compass and it even can't be drawn on geogebra (The point E comes outside $\overline{AC}$, while the question states that it is on the line)
Using Extension of Pythagoras Theorem ($a^2 = b^2 + c^2 + bc \sqrt 3$) where the values of the number rooted is correspondent to the angles or cosine rule, I get irrational numbers as the lengths and am not able to proceed further. I intended using Herons formula due to the lack of side lengths but am not able to find the length of $\overline{FE}$.
Can I get some hints to solve it using plain geometry(I know it has some thing to do with excentres but I'm not able to make any observations on this point)


It's $$\frac{6\cdot10\sin30^{\circ}}{2}=15.$$

The geometric solution.

Let $FK$ be an altitude of $\Delta FDE$.

Thus, since $$\measuredangle FDE=180^{\circ}-2\cdot75^{\circ}=30^{\circ},$$ we obtain: $$FK=\frac{1}{2}FD=3$$ and $$S_{\Delta DEF}=\frac{3\cdot10}{2}=15.$$

  • $\begingroup$ What formula did you use, and can you please tell me the geometric method of solving it if present $\endgroup$ – infixint943 Jun 25 '18 at 15:24
  • $\begingroup$ @Adithya Dsilva I added something. See now. $\endgroup$ – Michael Rozenberg Jun 25 '18 at 15:29

Your discovery that the problem is inconsistent is correct. $BFD$ and $CDE$ are $30-75-75$ triangles, which gives $BD=\frac 3{\sin 15^\circ}\approx 11.59, CD=\frac 5{\sin 15^\circ}\approx 19.31$ and $E$ lies beyond $A$ from $C$. Report that fact and you are done.

If you let $E$ be on the extension of $CA$ and want to find the area of $DEF$ anyway, note that $\angle FDE=30^\circ$, use the law of cosines to get the length of $FE$ and you have all three sides for Heron's formula.

  • $\begingroup$ How to find the area of it if it is outside the line? Can you please give me a hint or something in the current post? Thank You $\endgroup$ – infixint943 Jun 25 '18 at 15:23

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