What is the locus of $H$? Given a circle of radius $r$ and a chord $AB$ of length 1 unit, let $C$ be a variable point on the circle; let $I$ be the midpoint of $BC$ and let $H$ be the intersection of the line perpendicular to $AC$ from I. What is the locus of $H$?


Consider $AB$ as the reference axis for angles ($x$ axis), and consider $A$ as the origin. Let $\theta$ be the oriented angle between $AB$ and the (moving) line $AC$.
Let $H_1$ be the (moving) projection of $B$ onto  line $AC$.
Let $D$ be the (fixed) point such that $AD$ is a diameter of the given circle.
Let $\theta_0$ be the (fixed) oriented angle between $AB$ and $AD$.
The locus of $H_1$ is the circle with diameter $AB$ which has the following polar equation: 
$$r=r_1=\cos(\theta).$$
By definition, the locus of $C$ is the initial circle, with polar equation 
$$r=r_2=D_1 \cos(\theta-\theta_0).$$
for a certain fixed $D_1$ (the length of the circles' diameter, $AD$), and a certain fixed $\theta_0$ (which is the angle between $AB$ and $AD$).
$I$, being the midpoint of $H_1$ and $C$ (by Thales' theorem), has the following polar equation:
$$r=(r_1+r_2)/2 \ \ \Longrightarrow \ \ r=(D_1\cos(\theta-\theta_0)+cos (\theta))/2$$
which can be transformed (by expanding $\cos(a-b)=\cos a \sin b + ...$) into an expression: 
$$r=p cos(\theta)+q \sin(\theta)$$
which itself is known to be amenable (by taking $\delta:=\sqrt{p^2+q^2}$) to the form
$$r=\delta cos(\theta-\theta_1)$$
for a certain fixed angle $\theta_1$, which means that the locus is a certain circle passing through $A$ and, as one can verify immediately, passing as well through $B$ (for $\theta=0$). 
