## Challenge 374: Circle Challenge

A square S has vertices V_{1}, V_{2}, V_{3} and V_{4 }which lie on the circumference of a circle C with radius 1.

T is a straight line which is tangent to circle C.

The shortest distances from V_{1}, V_{2}, V_{3} and V_{4 }to T are d_{1}, d_{2}, d_{3}, and d_{4} respectively.

Show that the value of all of the following expressions remain constant, regardless of how T is positioned in relation to S.

(i) d_{1 }+ d_{2} +d_{3} + d_{4}

(ii) d_{1}^{2} + d_{2}^{2} +d_{3}^{2} + d_{4}^{2}

(ii) d_{1}^{3} + d_{2}^{3} +d_{3}^{3} + d_{4}^{3}