Understand the problem

Given a circle \Gamma, let P be a point in its interior, and let l be a line passing through P.  Construct with proof using a ruler and compass, all circles which pass through P, are tangent to \Gamma, and whose centres lie on l.

Source of the problem

RMO 2019 Maharashtra and Goa region

Topic
Geometry
Difficulty Level

Easy

Suggested Book

Tutorial

Try these problems first before watching the video or reading the hints:

(Send it to support@cheenta.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

Do you really need a hint? Try it first!

Consider an inversion with respect to a circle with centre P. Call this map f. Note that, given any point X, f(X) is constructible using ruler and compass. Construct the circle f(\Gamma).

Suppose \Gamma' is one of our solutions. Then f(\Gamma') is a line perpendicular to l=f(l) and tangent to f(\Gamma)

There can be no more than two lines perpendicular to l and tangent to f(\Gamma). Thus these two lines are the images of our solution circles.

Invert the lines back to get the solution circles.

Your content goes here. Edit or remove this text inline or in the module Content settings. You can also style every aspect of this content in the module Design settings and even apply custom CSS to this text in the module Advanced settings.

Connected Program at Cheenta

Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

Similar Problems

Sum of Sides of Triangle | PRMO-2018 | Problem No-17

Try this beautiful Problem on Geometry from PRMO -2018.You may use sequential hints to solve the problem.

Recursion Problem | AMC 10A, 2019| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-15, You may use sequential hints to solve the problem.

Roots of Polynomial | AMC 10A, 2019| Problem No 24

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-24, You may use sequential hints to solve the problem.

Set of Fractions | AMC 10A, 2015| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2015. Problem-15. You may use sequential hints to solve the problem.

Indian Olympiad Qualifier in Mathematics – IOQM

Due to COVID 19 Pandemic, the Maths Olympiad stages in India has changed. Here is the announcement published by HBCSE: Important Announcement [Updated:14-Sept-2020]The national Olympiad programme in mathematics culminating in the International Mathematical Olympiad...

Positive Integers and Quadrilateral | AMC 10A 2015 | Sum 24

Try this beautiful Problem on Rectangle and triangle from AMC 10A, 2015. Problem-24. You may use sequential hints to solve the problem.

Rectangular Piece of Paper | AMC 10A, 2014| Problem No 22

Try this beautiful Problem on Rectangle and triangle from AMC 10A, 2014. Problem-23. You may use sequential hints to solve the problem.

Probability in Marbles | AMC 10A, 2010| Problem No 23

Try this beautiful Problem on Probability from AMC 10A, 2010. Problem-23. You may use sequential hints to solve the problem.

Points on a circle | AMC 10A, 2010| Problem No 22

Try this beautiful Problem on Number theory based on Triangle and Circle from AMC 10A, 2010. Problem-22. You may use sequential hints to solve the problem.

Circle and Equilateral Triangle | AMC 10A, 2017| Problem No 22

Try this beautiful Problem on Triangle and Circle from AMC 10A, 2017. Problem-22. You may use sequential hints to solve the problem.