Let this point be M(m, m) and the slopes of the two outer tangents be k 1, then their equations are y-m = k 1 (x-m) and y-m = k2 (x-m).

Because the tangent is symmetrical about y=x, so

( 1-k 1)/( 1+k 1)=(k2- 1)/( 1+k2)

You can get K 1k2= 1.

The distance from the center O(5, 1) to two tangents k1x-y+m (1-k1) = 0 and k2x-y+m( 1-k2)=0 is the radius root number 2.

5k1-1+m (1-k1) = radical sign [2 (K 1 2+ 1)].

5k2- 1+m( 1-k2)= radical sign [2 (k2 2+1)]

Connect the equations and do the math yourself.