Understand the problem
Source of the problem
RMO 2019 Maharashtra and Goa region
Try these problems first before watching the video or reading the hints:
(Send it to email@example.com. Our priority response is for internal students, however we occasionally try to respond to external students as well). 1. How do you infer that a parallel line needs to be drawn through the center (to the given line AB (L)? 2. Can you find any isosceles triangle in the picture (once one of the little circles is drawn)? 3. How is the second small circle drawn?
Watch the video
Start with hints
Consider an inversion with respect to a circle with centre . Call this map . Note that, given any point , is constructible using ruler and compass. Construct the circle .
Suppose is one of our solutions. Then is a line perpendicular to and tangent to .
There can be no more than two lines perpendicular to and tangent to . Thus these two lines are the images of our solution circles.
Invert the lines back to get the solution circles.
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