There's a triangle ABC like on the image below triangle image where the lines coming from the corners divide the opposite edge into 3 equal parts. No other information is provided (so I'd not assume that there's a right angle anywhere, despite the image suggesting that)

How to prove that the area of the piece in the middle is $9/70$ of the whole triangle? I have trouble figuring out where to start and I'd appreciate any suggestions on what to use.

  • $\begingroup$ to start, consider that lines coming from each vertex divide the triangle into 3 triangles of equal areas $\endgroup$ – Vasya Jun 13 '18 at 12:44
  • $\begingroup$ @Vasya And then what? $\endgroup$ – Théophile Jun 13 '18 at 13:33

Given any polygon $P_1P_2\cdots P_n$, let $[P_1P_2\cdots P_n]$ be a shorthand for its area.

First, we need a lemma. For the configuration illustrated below, we have
$$\frac{[ABD]}{[ABC]} = \frac{uv}{u+v-uv} \quad\text{ where }\quad u = \frac{AE}{AC}, v = \frac{BF}{BC}$$ Lemma

To prove the lemma, let $a,b,c$ be the baricentric coordinates of $D$ with respect to $\triangle ABC$.
Notice $$\begin{cases} D \text{ lies on } AF &\implies b : c = 1-v : v\\ D \text{ lies on } BE &\implies a : c = 1-u : u \end{cases} \quad\implies\quad a : b : c = \frac1u -1 : \frac1v - 1 : 1$$ This leads to $$\frac{[ABD]}{[ABC]} = c = \frac{c}{a+b+c} = \frac{1}{\frac1u + \frac1v - 1} = \frac{uv}{u+v-uv} $$

Back to original question.

Let $\displaystyle\;\varphi(u,v) = \frac{uv}{u+v-uv}$ and label vertices as shown below.

Quadrilateral inside a triangle

Apply lemma to $\triangle APB, \triangle AQB, \triangle ARB, \triangle ASB$ with respect to $\triangle ABC$, we obtain

$$\frac{[APB]}{[ABC]} = \frac{[ARB]}{[ABC]} = \varphi\left(\frac13,\frac23 \right), \frac{[AQB]}{[ABC]} = \varphi\left(\frac23,\frac23\right) \quad\text{ and }\quad \frac{[ASB]}{[ABC]} = \varphi\left(\frac13,\frac13\right) $$

Notice $[PQRS] = [AQB] - [ARB] + [ASB] - [APB]$, the desired ratio is

$$\frac{[PQRS]}{[ABC]} = \varphi\left(\frac23,\frac23\right) + \varphi\left(\frac13,\frac13\right) - 2 \varphi\left(\frac13,\frac23 \right) = \frac12 + \frac15 - 2\cdot \frac27 = \frac{9}{70}$$


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