TITLE: $a^x+b^x=c^x$ in geometry
QUESTION [21 upvotes]: The Pythagorean theorem.
Let $A$, $C$, $B$ be three points on a line in this order, and let $D$ be another point, such that $\angle ADC =\angle CDB = 60^\circ$. Let $a=AD$, $b=BD$, $c=CD$. Then, $$a^{-1} + b^{-1} = c^{-1}.$$
Let $C_1$, $C_2$, $C_3$ be three circles that are tangent to each other and also tangent to a common line, such that $C_3$ lies between $C_1$ and $C_2$. Let $a$, $b$, $c$ be their respective radii. Then, $$a^{-1/2} + b^{-1/2} = c^{-1/2}.$$

See the figure below.
Are there any other results of this type in geometry?

REPLY [4 votes]: Here's another one. I don't think it's much popular, but the proof is not too terrible.

$$\large a^{2/3}+b^{2/3}=c^{2/3}$$