Understand the problem

Let \( \Gamma_1 \) and \( \Gamma_2 \) be two circles with unequal radii, with centers \(  O_1 \) and \( O_2 \) respectively, in the plane intersecting in two distinct points A and B. Assume that the center of each of the circles \( \Gamma_1 \) and \( \Gamma_2 \) are outside each other. The tangent to \( \Gamma_ 1 \) at B intersects \( \Gamma_2 \) again at C, different from B; the tangent to \(   \Gamma_2 \) at B intersects \(  \Gamma_1 \) again in D different from B. The bisectors of \( \angle DAB \) and \( \angle CAB \) meet \( \Gamma_1 \) and \( \Gamma_2 \) again in X and Y, respectively. different from A. Let P and Q be the circumcenters of the triangles ACD and XAY, respectively. Prove that PQ is perpendicular bisector of the line segment \( O_1 O_2 \). 

Tutorial Problems… try these before watching the video.

1. Suppose \( P O_1 Q O_2 \) be a kite (that is \( PO_1 = PO_2 \)  and \(  QO_1 1 = QO_2 \). Show that PQ is perpendicular bisector of the other diagonal $ O_1 O_2 $.$.

2. Show that for any two circles intersecting each other at two distinct points, the common chord is bisected perpendicularly by the line joining the center.

You may send solutions to support@cheenta.com. Though we usually look into internal students work, we will try to give you some feedback.

Now watch the discussion video

Subscribe to Cheenta’s youtube channel

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

likelihood and the moment graph

Restricted Maximum Likelihood Estimator |ISI MStat PSB 2012 Problem 9

This is a very beautiful sample problem from ISI MStat PSB 2012 Problem 9, It’s about restricted MLEs, how restricted MLEs are different from the unrestricted ones, if you miss delicacies you may miss the differences too . Try it! But be careful.

ISI MStat PSB 2013 Problem 2 | Application of sandwich Theorem

This is a very beautiful sample problem from ISI MStat PSB 2013 Problem 2 based on use of Sandwich Theorem . Let’s give it a try !!

ISI MStat PSB 2014 Problem 2 | Properties of a Function

This is a very beautiful sample problem from ISI MStat PSB 2014 Problem 2 based on the use and properties of a function. Let’s give it a try !!

exp

ISI MStat PSB 2012 Problem 3 | Finding the Distribution of a Random Variable

This is a very beautiful sample problem from ISI MStat PSB 2012 Problem 3 based on finding the distribution of a random variable . Let’s give it a try !!

ISI MStat Mock Test 1 | Cheenta Statistics Department

Mock Tests are important. They help you to analyze your own performance, by collecting your own data. Cheenta Statistics Department has prepared the open to all first National Mock Test for all the students preparing for ISI MStat 2020, during the COVID 19 situation....

ISI MStat PSB 2010 Problem 10 | Uniform Modified

This is a very elegant sample problem from ISI MStat PSB 2010 Problem 10, based on properties of uniform, and its behavior when modified. Try it!

vector space

ISI MStat PSB 2014 Problem 1 | Vector Space & Linear Transformation

This is a very beautiful sample problem from ISI MStat PSB 2014 Problem 1 based on Vector space and Eigen values and Eigen vectors . Let’s give it a try !!

ISI MStat PSB 2012 Problem 10 | MVUE Revisited

This is a very simple sample problem from ISI MStat PSB 2012 Problem 10. It’s a very basic problem but very important and regular problem for statistics students, using one of the most beautiful theorem in Point Estimation. Try it!

linear 1

ISI MStat PSB 2010 Problem 1 | Tricky Linear Algebra Question

This is a very beautiful sample problem from ISI MStat PSB 2010 Problem 1 based on Matrix multiplication and Eigenvalues and Eigenvectors.

ISI MStat PSB 2006 Problem 9 | Consistency and MVUE

This is a very simple sample problem from ISI MStat PSB 2006 Problem 9. It’s based on point estimation and finding consistent estimator and a minimum variance unbiased estimator and recognizing the subtle relation between the two types. Go for it!