Let AB be the diameter of a semicircle S. The | vedclass

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(C) Let $R$ be the radius of the semicircle $S$ and $r$ be the radius of the small circle with center $C$. Let $O$ be the center of the semicircle $S$

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a parabola

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(C) Let $R$ be the radius of the semicircle $S$ and $r$ be the radius of the small circle with center $C$. Let $O$ be the center of the semicircle $S$. The distance from $C$ to the diameter $AB$ is $r$. The distance from $C$ to the center $O$ is $R - r$ (since the small circle is tangent to the semicircle $S$). Let $T$ be the projection of $C$ onto the diameter $AB$. Then $CT = r$. Thus,the distance of $C$ from the point $O$ is equal to the distance of $C$ from the line $AB$ plus some constant,or more precisely,$CO + CT = R$. Since the distance from $C$ to the point $O$ is equal to the distance from $C$ to a line parallel to $AB$ at a distance $r$ below it,this definition fits the geometric property of a parabola where the distance to a fixed point (focus) equals the distance to a fixed line (directrix). Therefore,the locus of $C$ is an arc of a parabola.

Let $AB$ be the diameter of a semicircle $S$. The locus of the centres of circles which are tangent to $AB$ and to $S$ is an arc of

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