Kyushu University 2020: Cutting a Regular Tetrahedron

Problem

Find the maximum value of the cross-sectional area when a tetrahedron OABC with vertices at O ${(0, 0, 0)}$, A ${(1, 1, 0)}$, B ${(1, 0, 1)}$, and C ${(0, 1, 1)}$ in coordinate space is cut by a plane perpendicular to the z-axis.

In the original problem, part (1) asks to determine the values of ${p, q, r, s}$ by setting B as ${(1, 0, p)}$ and C as ${(q, r, s)}$ and treating it as a regular tetrahedron.
To find the cross-sectional area, it is likely impossible to calculate without drawing a figure to grasp the situation.
When cut by the plane ${z=k}$,
The coordinates of the intersection with edge AB are P${(1, 1-k, k)}$
The coordinates of the intersection with edge AC are Q${(1-k, 1, k)}$
The coordinates of the intersection with edge OB are R${(k, 0, k)}$
The coordinates of the intersection with edge OC are S${(0, k, k)}$
If we define these, the cross-section becomes a rectangle PQRS. Then, the area becomes a quadratic function of ${k}$, and the maximum value can be calculated as ${\dfrac{1}{2}}$.

Visualizing spatial figures is quite difficult. Let's try looking at this while changing the perspective or moving the plane ${z=k}$.

Clicking the link will take you to the following screen.

Lined up on the left are circular sliders. Dragging the green point rotates the figure around the coordinate axes. In the initial figure, OABC does not look like a regular tetrahedron, but it looks like one when rotated. It is easier to grasp the image if you rotate it yourself while looking at it.

While looking at this figure, you are supposed to imagine the cross-section when cut by the plane ${z=t}$, but that is quite difficult. Pressing the "Cut" button will display the cross-section.

Let's try changing the height of the plane (the value of ${t}$) using the slider on the left.

Let's try rotating it with the circular slider. This is what it looks like when viewed from directly above (from the positive side of the z-axis).

This is the figure necessary to derive the answer. If you can imagine this figure without such tools, you can solve the problem.

*The figures were created using Cinderella (CindyJS).