Can you draw this tangent ?

Geometry Level 3

You are given a white sheet, a ruler without any graduation, and a compass. Drawn on the sheet are a circle C , C, its center O , O, and a point A A on the circle. Since you can only draw circles and straight lines, what is the fewest number of circles and/or straight lines you have to draw in order to have drawn the tangent line of the circle that passes through the point A ? A?

1 2 3 4 5 6 7 It is impossible

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2 solutions

Michael Mendrin
Aug 12, 2018

First draw an arbitrary red circle with center on the circumference of black circle and passing through point A A . Then draw blue circle, and then draw purple tangent.

The dashed line and circle shows the symmetry

You can draw it with only two circles and a straight line :

Let B B be any other point on the circle C C ( B B must not be on the same diameter as A A ), then draw the circle of center B B and radius B A BA , let it be circle C 2 C2 . Let the second intersection of C C and C 2 C2 be P P .

Then draw the circle C 3 C3 of center A A and radius A P AP , let the second intersection of C 2 C2 and C 3 C3 be T T .

Finally draw the straight line ( A T ) (AT) , you have just drawn the tangent line of C C that passes through the point A.

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