<s>[INST] Intersection of perpendicular chords: is this true for a sphere? It is true that in a circle with radius $R$, if the intersection of any two perpendicular chords divides one chord into lengths $a$ and $b$ and divides the other chord into lengths $c$ and $d$, then
$$a^2 + b^2 + c^2 + d^2 = 4R^2.$$
Does this equation hold for a sphere with radius $R$, and three mutually perpendicular chords:
$$a^2 + b^2 + c^2 + d^2 + e^2 + f^2 = 6R^2?$$
 [/INST] For circles:
From the circle center draw perpendicular lines to the chords. Let the intersection of the lines with the chords be $X$ and $Y$, respectively. Let $|OX|=x$, $|OY|=y$.

We have (see figure):
$$\begin{align}
a&=\sqrt{R^2-x^2}+y,\\
b&=\sqrt{R^2-x^2}-y,\\
c&=\sqrt{R^2-y^2}-x,\\
d&=\sqrt{R^2-y^2}+x,
\end{align}$$
so that
$$
a^2+b^2+c^2+d^2=4R^2,
$$
as claimed.
Try to apply the same construction for the sphere case.

 Drawing from the center of the sphere the planes perpendicular to the chords and denoting the intersection points $X,Y,Z$ you will obtain: $$a^2+b^2+c^2+d^2+e^2+f^2=6R^2-2(x^2+y^2+z^2),$$ so that the claim does not hold for a sphere.

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