Conjecture & Proof - Eyeball to Eyeball
Hint: The proof rests on similarity - try drawing in the radii at right angles to the tangent lines. Consider the similar triangles that can be found in the diagrams in order to create expressions for the length of the chords.
Celestial Sphere with some constellations (very crude)
Investigation 1: Is AA a Similarity Shortcut?
a squard to savared equals squared