Minimum force required to rotate a lamina

[haruspex]

how do we conclude that magnitude of friction is constant

The weight per unit area is constant. When the lamina moves, all parts go into kinetic friction, so when it is about to move all parts must be at max static friction.

 

[PhysicsBoi1908]

Ah! Thank you very much.

 

[etotheipi]

I see a way.
Drop a perpendicular from C to meet AB at D. Consider the torques required to rotate the two smaller triangles about C.

It is very clever! If you don't explicitly cut the lamina there will be an internal force between the two parts. The result is that the torque you have to apply at A/B is slightly more than calculated and the torque you have to apply at B/A is slightly less than expected. But these two discrepancies will cancel exactly, and their sum will just be the same as if the lamina were not actually connected, except at the hinge (i.e. no internal force between the two parts).