I'm a little too lazy to dig out a book, so let's see if we can think our way through the problem without getting too deep into electrostatics.
Suppose your initial capacitor had some kind of conductive coating on the material with the high dielectric constant. It wouldn't be electrically connected to anything, it would just be there, and it would float at whatever voltage would have existed in free space right at the dielectric/airgap interface had it not been there in the first place. You could call it an absentee plate: there's no way you would know it's there from the usual electical impedence tests you could do on the two outside plates.
If you can buy that image (it IS a valid model) then the question becomes one of having not of one cap with two different materials in it, but two caps in series, one with an air gap of d/2 -- that's the one without the high dielectric material in it -- and another with a different effective air gap because the the higher dielectric constant material.
Would you agree you could in this analysis substitute an air gap cap for the one with the higher dielectric constant material in the gap? Sure. The numbers work out that a airgap of d/2/K will have the same capacitance as one with a d/2 gap filled with material of a dielectric constant K.
If you could understand the equivelences I suggested here, you know two things. The more trivial one is how to calculate the overall capicatance of the circuit, but the better one is a way of thinking about modeling some of these things.
Or not.
This is worth exactly what you paid for it, but it was fun thinking about how to solve the problem. My brain worked on something I might have learned an Ohmygod 40some years ago!