Is this quadrilateral tangential?
MATL, 11 10 bytes

5:)d|2e!sd


Input is a vector of four complex numbers. Output is 0 (which is falsy) if tangential, or nonzero (which is truthy) if not tangential.

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Explanation

The code computes the difference between sums of lengths of opposite sides. This difference is zero if and only if the quatrilateral is tangential.

5:   % Range [1 2 3 4 5]
)    % Implicit input: complex vector of length 4. Index into it modularly.
     % This repeats the first vertex after the last
d    % Consecutive differences
|    % Absolute value, element-wise
2e   % Reshape as a 2-column matrix, in column-major order
!    % Transpose
s    % Sum of each column. Gives a vector of length 2
d    % Consecutive difference