Construction of segment of given length through an intersection of two circles

circlesgeometric-constructiongeometry

"Through an intersection point of two circles, draw a secant such that its segment inside the given disks is congruent to a given length.

Hint: Construct a right triangle whose hypotenuse is the segment between the centers of the given disks,
and one of the legs is congruent to a half of the given length".

enter image description here
Let's say segment EF is given and we have to construct secant through point K such that its length inside disks A and B equals segment EF.

Many thanks for any ideas or suggestions.

Best Answer

Hint:

enter image description here

$ACSR$ is a rectangle. Do you see why $AC=RS=\frac12 LM$?

Solution: construct - as was suggested - a right triangle $ABC$ whose hypotenuse $AB$ is the segment between the centers of the given disks, and one of the legs ($AC$) is congruent to $\frac12EF$. Draw through the point $K$ a line parallel to $AC$. It will intersect the circles in the points $L$ and $M$. Then $LM=EF$. The construction is possible only if $\frac12EF<AB$, in which case there are two solutions according to the number of possibilities to construct the triangle $ABC$.

Related Question