Let $AB$ be a line segment. Consider a circle $\mathfrak C$ with center at $B$ and of radius $1$. Let $Ax$ be a line perpendicular to $AB$ (with a right angle $BAx$), and $O$ be a point on $Ax$. Suppose the line $OB$ intersects $\mathfrak C$ in $D$ (say, firstly). Using straightedge and compass find the position of such $O$ with $OA=OD$.
If we extend $DO$ from $O$ to reach a point $E$ with $OE=OD$, then the circle passing through $A, D, E$ is tangent with $AB$ at $A$, and we have $AB^2=BD. BE=BE$.
Let $ABC$ be a right triangle where the angle $B$ is the right angle.
It is well known that the square of the altitude $BH$ is the product of the segments $AH$ and $HC$.
Thus if $AH=1$ and $BH=x$, we have $HC=x^2.$
We can start with the triangle $AHB$ and complete the structure of $ABC$ by extending $AH$ to $AC$, where $C$ is the intersection of $AC$ and $BC$