Let $P_1 P_2 P_3 \dotsb P_{10}$ be a regular polygon inscribed in a circle with radius $1.$ Compute
\[(P_1 P_2)^2 + (P_1 P_3)^2 + (P_1 P_4)^2 + \dots + (P_{8} P_{9})^2 + (P_{8} P_{10})^2 + (P_{9} P_{10})^2.\](The sum includes all terms of the form $(P_i P_j)^2,$ where $1 \le i < j \le 10.$ We write $P_iP_j$ to mean the length of segment $\overline{P_iP_j}$.)

Maybe use sum of square lengths and the height of the 10 isoceles triangles?