# Interpretation issues with a problem George Polya?

I have this problem from the textbook "How to solve it

10 The vertex of a pyramid opposite the base is called the apex.

(a) Let us call a pyramid “isosceles” if its apex is at the same distance from all vertices of the base. Adopting this definition, prove that the base of an isosceles pyramid is inscribed in a circle the center of which is the foot of the pyramid’s altitude.

(b) Now let us call a pyramid “isosceles” if its apex is at the same (perpendicular) distance from all sides of the base. Adopting this definition (different from the foregoing) prove that the base of an isosceles pyramid is circumscribed about a circle the center of which is the foot of the pyramid’s altitude

I could understand the item a) its easy to understand because we can visualize the distance between the points.

But for the item B) How is the appex vertex (is at the same distance than the sides of the base) because the side of the base is a whole linear, shouldn't it says to the middle point of the side of the base, or what?

This is the solution:

1. The base of the pyramid is a polygon with n sides. In the case (a) the n lateral edges of the pyramid are equal; in the case (b) the altitudes (drawn from the apex) of its n lateral faces are equal. If we draw the altitude of the pyramid and join its foot to the n vertices of the base in the case (a), but to the feet of the altitudes of the n lateral faces in the case (b), we obtain, in both cases, n right triangles of which the altitude (of the pyramid) is a common side: I say that these n right triangles are congruent. In fact the hypotenuse [a lateral edge in the case (a), a lateral altitude in the case (b)] is of the same length in each, according to the definitions laid down in the proposed problem; we have just mentioned that another side (the altitude of the pyramid) and an angle (the right angle) are common to all. In the n congruent triangles the third sides must also be equal; they are drawn from the same point (the foot of the altitude) in the same plane (the base): they form n radii of a circle which is circumscribed about, or inscribed into, the base of the pyramid, in the cases (a) and (b), respectively. [In the case (b) it remains to show, however, that the n radii mentioned are perpendicular to the respective sides of the base; this follows from a well-known theorem of solid geometry on projections.] It is most remarkable that a plane figure, the isosceles triangle, may have two different analogues in solid geometry

Another item from the solution in this case that I couldn't get was from

In fact the hypotenuse [a lateral edge in the case (a), a lateral altitude in the case (b)]

Whats mean that the lateral altitude is the hipotenuse, should be either the lateral edge?

Thank you

• It says, the same perpendicular distance from all sides of the base. That means, you drop a perpendicular from the apex to each side of the base, and each of those perpendiculars has the same length. Commented Sep 19, 2022 at 2:43
• @GerryMyerson Hello Gerry, but that definition is not implied also in (a), because, if we say that the lateral vertex are in the same position from the apex isn't that equal that the perpendicular from the apex is the same lenght for all of the sides like in (b)? Commented Sep 19, 2022 at 20:46
• Sorry, I don't understand your comment. Commented Sep 20, 2022 at 3:39