# Construct tangent to a circle

Using a ruler and a compass how can construct a line through a point and tangent to a circle. What I don't want is to eyeball the line by trying to line-up the ruler over the circle. Best if I could construct the point of intersection first and then draw the line.

PS. I know how to do it mathematically, I just don't know the steps for geometry, given A, C and the circle to find D.

Update Based on answers here is the constructors. Thanks for the quick responses.

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You could look at mathopenref.com/consttangents.html – Ross Millikan Oct 29 '12 at 17:39

Draw a Thales circle over the segment $AC$, it will intersect the desired $D$, because $AD\perp DC$:

1. Draw the segment $AC$.
2. Construct its midpoint $F$.
3. Draw a circle with origin $F$ and radio $FA(=FC)$.
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Intersect the circle having $AC$ as a diameter with the initial circle: you will find the two points $D,D'$ such that $CD$ and $CD'$ are tangent to the initial circle. This comes from the fact that the circle is the locus of points that "see" any diameter under an angle equal to $\frac{\pi}{2}$.

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1. From AC, find its midpoint F.
2. Draw the circle using F as center and FA (or FC) as radius.
3. The point(s) of intersection of the circles is D.
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Thanks, but how is this answer different from the one Berci provided? – ja72 Aug 12 '13 at 17:30
@ja72, Yes but have the extra description cut. – Mick Aug 22 '13 at 12:56