Kennedy’s theorem (instantaneous centres): According to Kennedy’s theorem, for three bodies having plane motion relative to one another, the three instantaneous centres corresponding to each pair lie on what geometric locus?

Introduction / Context:
Kennedy’s theorem (also called Aronhold–Kennedy) is a key kinematic principle for mechanisms. It states a collinearity property of instantaneous centres (ICs), which greatly simplifies velocity analysis in linkages such as the four-bar, slider-crank, and geared systems.

Given Data / Assumptions:

• Three rigid bodies in plane motion with respect to each other.
• Instantaneous centre defined for each pair at a given instant.
• Rigid body assumption; small-time instant where ICs exist.

Concept / Approach:
If the three bodies are numbered 1, 2, and 3, their pairwise ICs are I₁₂, I₂₃, and I₃₁. Kennedy’s theorem states that these three ICs are collinear at any instant. This enables finding an unknown IC by joining the other two and locating their intersection with mechanism geometry.

Step-by-Step Solution:

Verification / Alternative check:
Velocity polygons or relative velocity equations can be used to confirm collinearity since the velocity of a point is perpendicular to the line from the IC to that point, leading to the same straight-line condition.

Why Other Options Are Wrong:

• Point, two lines, triangle, circle: none captures the specific collinearity relation among the three ICs.

Common Pitfalls:
Confusing fixed vs. moving ICs; misidentifying ICs at slider contacts; assuming ICs form a triangle (they do not—only the links do).

Final Answer:
A straight line