Geometric Mean & Right Triangles

Instructions

Step 1: Drop a perpendicular from the vertex Z. Step 2: Show the product of the newly formed segments (a and b). Step 3: Take the square root of that product. Step 4: Reveal the measurement of the height of the right triangle. Step 5: Drag vertex X, Y, and/or Z.

Conjecture

What is going on here? How can we prove it?