A regular quadrilateral pyramid has a base $ABCD$ and apex $E$, with the center of the base being $O$. The length of the side edge is twice the length of the base edge. Where on the segment $EO$ is the point $M$ such that the sum of the areas of triangles $MAE$ and $MAB$ is minimized?
I. solution. Let's choose the base edge's half as the unit of length, let the height of the pyramid be $O E=m$ (for now, ignoring the given ratio $A E$ : $A B=2$) and $O M=x(>0)$.

![](https://cdn.mathpix.com/cropped/2024_05_02_b6ef0e0f939d166c7166g-1.jpg?height=582&width=618&top_left_y=262&top_left_x=735)

Figure 1

Thus, denoting the midpoint of edge $A B$ by $F$, we have $F M=\sqrt{x^{2}+1}$, on the other hand $A O=\sqrt{2}$, and the area sum to be examined (Figure 1)

$$
t(x)=A F \cdot F M+\frac{1}{2} M E \cdot A O=\sqrt{x^{2}+1}+\frac{m-x}{\sqrt{2}}
$$

if $01$, then $t^{\prime}(x)>0$. Furthermore, when $A E=4$, $m=\sqrt{14}>1$, according to these, the area sum has its minimum at $x=1$.

II. solution. Transform the first expression of $t(x)$ as follows:

$$
t(x)=A F\left(F M+\frac{M E}{\sqrt{2}}\right)
$$

Since $A F$ is constant, according to this, we need to choose $x$ so that the sum of the two segments in parentheses is as small as possible. We can give a visual meaning to this problem in the plane section $F O E$ of the pyramid. Draw a half-line $E G$ through $E$ that forms a $45^{\circ}$ angle with the height $E O$ on the side not containing $F$, and let the projection of $M$ on this be $M^{\prime}$ (Figure 2).

![](https://cdn.mathpix.com/cropped/2024_05_02_b6ef0e0f939d166c7166g-1.jpg?height=499&width=480&top_left_y=1992&top_left_x=801)

Figure 2

The second term in the parentheses is clearly $M M^{\prime}$, and the sum of the two terms is the distance we travel from $F$ to $E G$, if we go straight to $E O$ and then continue straight to $E G$, moving perpendicularly to $E G$ in the latter part. This path is obviously the shortest when we do not change direction when crossing $O E$, i.e., when $\angle O F M=45^{\circ}$, which means $O M=O F=1$.

Remark. The method used in the latter solution can be effectively applied to the following problem as well. Given a line $e$ in the plane, a point $B$ on it, and a point $A$ outside it. Determine the point $P$ of $e$ for which the

$$
\lambda \cdot A P+\mu \cdot P B
$$

sum is minimal, where $\lambda, \mu$ are arbitrary positive numbers.