$a^x+b^x=c^x$ in geometry

Question

• 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?

Answer 1

Here's another one. Let $ABCD$ be a trapezoid of area $c$ and let $O$ be the intersection point of its diagonals. If $a$ and $b$ are areas of triangles $AOD$ and $BOC$ (or $AOB$ and $COD$) then $$a^{1/2}+b^{1/2}=c^{1/2}.$$

Answer 2

Exponents $1,2, -1, -2,1/2$, and $-1/2$ are probably the most common, with the complexity of the implied polynomial equation $P(a,b,c)=0$ increasing in that order. As a random example, the height in the crossed ladders problem satisfies $h^{-1} = a^{-1} + b^{-1}$. For the altitude of a right triangle, $h^{-2} = a^{-2} + b^{-2}$. There must be many more.

Exponent $2/3$: the envelope of a line segment of length $L$ with endpoints on the $x$ and $y$ axes is $x^{2/3} + y^{2/3} = L^{2/3}$. The curve is called astroid.

Answer 3

Here's another one. I don't think it's much popular, but the proof is not too terrible.

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