We accept as proven the following two statements. Let $P$ be any interior point of triangle $ABC$, and let the rays $AP$, $BP$, $CP$ intersect the sides $BC$, $CA$, $AB$ at points $A_1$, $B_1$, $C_1$ respectively. Then the sum $PA + PB + PC = s$ is less than the sum of the two largest sides of the triangle, and the sum $PA_1 + PB_1 + PC_1 = s_1$ is less than the largest side of the triangle. (This was a problem from the 1973 Dániel Arany competition.)

Now let $ABC$ be an equilateral triangle, and choose the length of its side to be one unit. Which numbers $c$ are such that there exists a point $P$ for which $s > 2c$ and $s_1 > c$?
According to the accepted statements, in our case $s2 c$ and $s_{1}>c$.

![](https://cdn.mathpix.com/cropped/2024_05_02_594d00dd98f954c651d6g-1.jpg?height=591&width=643&top_left_y=260&top_left_x=730)

If we have found a point $P$ for a number $c_{0}$, this point obviously satisfies every smaller $c$ number, that is, $c0$ shows a deficiency with respect to 1, i.e., $1-c_{0}=\varepsilon, c_{0}=1-\varepsilon$.

Measure a $d$ such that $c_{0} < s_{1} > c_{0}$. Based on the three symmetry axes of the triangle $ABC$, it is sufficient to focus on the values $d \leqq \frac{1}{2}$. 1.) In the triangle $A_{1} A C$, by the triangle inequality, $A_{1} A > A C - A_{1} C = 1 - d$. The triangles $A_{1} A C$ and $B_{1} B A$ are congruent, and the triangle $B_{1} A P^{*}$ is similar to them because their angles are pairwise equal, thus

$$
\frac{A P^{*}}{A C} = \frac{A B_{1}}{A A_{1}} = \frac{d}{A A_{1}} \quad P^{*} A_{1} = A A_{1} - A P^{*} > (1 - d) - A P^{*} > (1 - d) - 2d = 1 - 3d
$$

Therefore, for $s_{1} > c_{0} = 1 - \varepsilon$ to hold, it is sufficient if $1 - 3d > 1 - \varepsilon$, i.e., $d < \frac{\varepsilon}{3}$. - For $P^{*} B > 1 - 2d$, and since $P^{*} B > 1 - 2d$, thus $P^{*} B + P^{*} C > 2 - 4d$, so $\frac{s}{2} > 1 - 2d > c_{0} = 1 - \varepsilon$ to hold, it is sufficient if $d < \frac{\varepsilon}{2}$. - In the case of $s_{1}$, we obtained a stronger restriction on $d$ than before, so if we mark $A_{1}$ and $B_{1}$ with a $d$ smaller than $\frac{\varepsilon}{2}$, then both conditions are satisfied for the above $P^{*}$, and $P^{*}$ meets the requirement for the desired $P$. - This proves our statement, the sought $c$ numbers are those for which $c < 1$.

Remarks. 1. We made very rough approximations in our estimates - of course always in the appropriate direction - but for our purpose, this was sufficient.

2. It was not necessary to prove that in an equilateral triangle, for any internal point $P$, the segments $A A_{1}, B B_{1}, C C_{1}$ are smaller than 1.