Converse of Geometric mean theorem [closed]
The conjecture :
Let $\triangle ABC$ with $C=90^∘$, and let $D∈[AB]$. If $CD^2=AD⋅DB$, then $CD$ is the altitude.
is false. The simplest counterexample is : let $D$ be the midpoint of hypotenuse $AB$. Then $D$ is center of circle through $A,B,C$. So that $CD=AD=BD=$ radius of this (circum)circle.
Indeed $CD^2=AD\cdot DB$, but $CD$ is not the altitude (except for the special case of isosceles right triangle).