## Task 1 - 211211

Given the side length $a$ of an equilateral triangle $A B C$. On this triangle as the base, a pyramid $A B C D$ with the following properties (1), (2) is to be constructed:

(1) The edges $A D, B D$ and $C D$ have equal length $s$.

(2) The height $h$ of the pyramid is the arithmetic mean of $a$ and $s$.

Investigate whether such a pyramid $A B C D$ exists and whether $h$ and $s$ are uniquely determined by $a$. If this is the case, determine $h$ and $s$.
I. Suppose a pyramid $ABCD$ has the required properties. If $F$ is the foot of the perpendicular from $D$ to the plane $\epsilon$ through $A, B, C$, then by the Pythagorean theorem (1) we have

$$
F A = F B = F C = \sqrt{s^2 - h^2}
$$

Thus, $F$ is the circumcenter of the equilateral triangle $ABC$ and, therefore, also the orthocenter and centroid of this triangle. The height of the triangle $ABC$ is $\frac{1}{2} a \sqrt{3}$, and the centroid divides the medians in the ratio $1:2$, so we have $F A = F B = F C = \frac{1}{3} a \sqrt{3}$. Therefore,

$$
s^2 - h^2 = F A^2 = \frac{1}{3} a^2
$$

From (2) it follows that

$$
h = \frac{1}{2}(h + s)
$$

Substituting this into (3), we get

$$
\begin{aligned}
\frac{1}{3} a^2 & = s^2 - \frac{1}{4} a^2 - \frac{1}{2} a s - \frac{1}{4} s^2 \\
0 & = s^2 - \frac{2}{3} a s - \frac{7}{9} a^2 \\
s_{1,2} & = \frac{a}{3}(1 \pm 2 \sqrt{2})
\end{aligned}
$$

Since $s > 0$, we have

$$
s = \frac{a}{3}(1 + 2 \sqrt{2}) \quad ; \quad a + s = \frac{a}{3}(4 + 2 \sqrt{2})
$$

From this, using (4), we get

$$
h = \frac{a}{3}(2 + \sqrt{2})
$$

Therefore, a pyramid $ABCD$ can only have the required properties if $D$ lies on the perpendicular erected at the centroid $F$ of the triangle $ABC$ in the plane $\epsilon$ such that

$$
F D = \frac{a}{3}(2 + \sqrt{2})
$$

II. For any such pyramid $ABCD$, we indeed have

$$
\begin{aligned}
A D = B D = C D & = \sqrt{F A^2 + F D^2} = \sqrt{\frac{1}{3} a^2 + \frac{1}{9} a^2 (2 + \sqrt{2})^2} \\
& = \frac{a}{3} \sqrt{3 + 4 + 4 \sqrt{2} + 2} \\
& = \frac{a}{3}(1 + 2 \sqrt{2})
\end{aligned}
$$

Thus, (1) is satisfied with $s = \frac{a}{3}(1 + 2 \sqrt{2})$, and the height $F D = \frac{a}{3}(2 + \sqrt{2})$ is also equal to the arithmetic mean $\frac{1}{2}(a + s)$, so (2) is satisfied.

Therefore, it is shown: There exists a pyramid $ABCD$ with the properties (1), (2); in this case, $h$ and $s$ are uniquely determined, and specifically, $h = \frac{a}{3}(2 + \sqrt{2})$, $s = \frac{a}{3}(1 + 2 \sqrt{2})$.

Adapted from $[5]$