<s>[INST] Sum of squares of ratios of diagonal of a regular heptagon A problem from the 1998 Lower Michigan Math Competition...
A regular heptagon has diagonals of two different lengths.  Let $a$ be the length of a side, $b$ the length of a shorter diagonal, and $c$ the length of a longer diagonal.  Prove that
$$
\frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2} = 6 \quad \text{ and } \quad \frac{b^2}{a^2} + \frac{a^2}{c^2} + \frac{c^2}{b^2} = 5.
$$
What I have so far:

*

*Introduce coordinates so that the heptagon consists of the seventh roots of unity in the complex plane.

*Let $\omega = e^{2 \pi i/7}$ so that the vertices are $1, \omega, \omega^2, \dots, \omega^6$.  Then $a = |1-\omega|$ and $b = |1-\omega^2| = |1-\omega||1+\omega|$ and $c = |1-\omega^3| = |1-\omega||1+\omega+\omega^2|$.

*The first requested expression then becomes
$$
\frac{1}{|1+\omega|^2} + \frac{|1+\omega|^2}{|1+\omega+\omega^2|^2} + |1+\omega+\omega^2|^2.
$$

*All those modulus-squared expressions can be written as (thingy) times (conjugate of thingy), and the conjugate of a power of $\omega$ is another power of $\omega$.  For example,
$$
|1+\omega|^2 = (1+ \omega)(1 + \overline{\omega}) = (1 + \omega)(1 + \omega^6).
$$
Using this, the first requested expression becomes
$$
\frac{1}{(1+\omega)(1+\omega^6)} + \frac{(1+\omega)(1+\omega^6)}{(1+\omega+\omega^2)(1+\omega^5+\omega^6)} + (1 + \omega + \omega^2)(1 + \omega^5 + \omega^6).
$$
Annnnd...that's about where I ran out of steam.  Not sure how much of this is progress, but I really want the complex-roots-of-unity approach to pan out!
 [/INST] Adding details to David's answer---too much algebra to fit in a comment.  ;-)
Some milestones along the way:

*

*Taking the modulus-squared on both sides of David's expression for $\frac{1}{1+\omega}$ yields
$$
\frac{1}{|1+\omega|^2} = |1+\omega^2 + \omega^4|^2
= (1+\omega^2+\omega^4) \overline{(1+\omega^2+\omega^4)}
= (1+\omega^2+\omega^4)(1+\omega^5+\omega^3)
$$

*Expanding the polynomial and reducing powers of $\omega$ modulo 7 yields
$$
\frac{1}{|1+\omega|^2} = 1 + \omega^5 + \omega^3 + \omega^2 + \omega^7 + \omega^5 + \omega^4 + \omega^9 + \omega^7 = 3 + 2 \omega^2 + \omega^3 + \omega^4 + 2 \omega^5
$$

*Likewise,
$$
|1+\omega+\omega^2| = (1+\omega+\omega^2)(1+\omega^6+\omega^5) = \cdots = 3 + 2 \omega + \omega^2 + \omega^5 + 2 \omega^6
$$

*Multiplying both sides of David's expression for $\frac{1}{1+\omega+\omega^2}$ by $1+\omega$ yields
$$
\frac{1+\omega}{1+\omega+\omega^2} = - \omega(1+\omega)(1+\omega^3)
$$
and taking modulus-squared on both sides,
$$
\left|\frac{1+\omega}{1+\omega+\omega^2}\right|^2 = (1 + \omega)(1 + \omega^6)(1+\omega^3)(1+\omega^4) = (2 + \omega + \omega^6)(2 + \omega^3 + \omega^4) = \cdots = 4 + 2 \omega + \omega^2 + 3 \omega^3 +  3\omega^4 + \omega^5 + 2 \omega^6.
$$
-- Adding the three equations together,
$$
\frac{1}{|1+\omega|^2} + |1+\omega+\omega^2|^2 + \left|\frac{1+\omega}{1+\omega+\omega^2}\right|^2 = 10 + 4 \omega + 4 \omega^2 + 4 \omega^3 + 4 \omega^4 + 4 \omega^5 + 4 \omega^6.
$$

*But that latter expression is
$$
6 + 4 (1 + \omega + \omega^2 + \cdots + \omega^6) = 6 + 4(0) = 6.
$$
</s>