WLOG take the arc endpoints to be (sin(\theta_1), 1 - cos(\theta_1)) and (sin(\theta_2), 1 - cos(\theta_2)), for 0 < \theta_1 < \theta_2 < \tau/4. The arc length is (\theta_2 - \theta_1).

The length of the other two sides
= (sin(\theta_2) - sin(\theta_1)) + ((1 - cos(\theta_2)) - (1 - cos(\theta_1)))
= sin(\theta_2) - sin(\theta_1) + cos(\theta_1) - cos(\theta_2)
= (sin-cos)(\theta_2) - (sin-cos)(\theta_1)
> (\theta_2 - \theta_1), by the fact that the (sin-cos) function always has derivative > 1 in the range (0, \tau/4).