Given that X is the incentre of triangle ABY,
we have ∠BAX = ∠XAY.
Therefore, ∠BDC = ∠BAC = ∠BAX = ∠XAY = ∠XDY = ∠BDY.
This shows that C, D, Y are collinear.
Therefore, ∠CYX + ∠XYD = 180°.
But the left-hand side equals (180° - ∠CBD) + (180° - ∠CAD).
Since ∠CBD = ∠CAD,
we obtain 180° = 360° - 2∠CAD.
This shows that ∠CAD = 90°