Understand the problem

Let $ABC$ be a triangle, $O$ its circumcenter, $S$ its centroid, and $H$ its orthocenter. Denote by $A_1, B_1$, and $C_1$ the centers of the circles circumscribed about the triangles $CHB, CHA$, and $AHB$, respectively. Prove that the triangle $ABC$ is congruent to the triangle $A_1B_1C_1$ and that the nine-point circle of $\triangle ABC$ is also the nine-point circle of $\triangle A_1B_1C_1$.

Source of the problem
IMO longlist 1992
Topic
Geometry
Difficulty Level
Hard
Suggested Book
Challenge and Thrill of Pre-college Mathematics

Start with hints

Do you really need a hint? Try it first!

Prove that the reflections of H with respect to the sides AB,BC,CA all lie on the circumcircle of ABC.
Study the reflection of \odot AHB with respect to AB.
Combining the previous hints, show that A_1,B_1, C_1 are just reflections of O on the sides of ABC.

From the last hint, we have BA_1=BO=R=CO=CA_1 where R is the circumradius of ABC. Hence,  BOCA_1 is a rhombus and A_1C||BO. Similarly, BO||AC_1 hence A_1C||AC_1. As CAC_1A_1 is a parallelogram, we also have A_1C_1=AC. We can similarly prove that A_1B_1=AB and B_1C_1=BC. Thus ABC\cong A_1B_1C_1. This implies that it suffices to show that the centres of the two nine-point circles coincide. Remember that the nine-point centre is the midpoint of the line joining the circumcentre and the orthocentre. Claim  H is the circumcentre of A_1B_1C_1.   Proof  Note that A_1H=R (as A_1 is the centre of \odot BHC and \odot BHC is a reflection of \odot ABC. Similarly, B_1H=C_1H=R.     Claim  O is the orthocentre of A_1B_1C_1.   Proof   As A_1 is the reflection of O on BC, A_1O\perp BC. As BC\parallel B_1C_1, A_1O\perp B_1C_1. Similarly, B_1O\perp C_1A_1 and C_1O\perp A_1B_1. Thus O is the orthocentre of A_1B_1C_1.  

Thus the centres of the nine-point circles of ABC and A_1B_1C_1 coincide.

Watch the video (Coming Soon)

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

GCF & Rectangle | AMC 10A, 2016| Problem No 19

Try this beautiful Problem on Geometry on Rectangle from AMC 10A, 2010. Problem-19. You may use sequential hints to solve the problem.

Fly trapped inside cubical box | AMC 10A, 2010| Problem No 20

Try this beautiful Problem on Geometry on cube from AMC 10A, 2010. Problem-20. You may use sequential hints to solve the problem.

Measure of angle | AMC 10A, 2019| Problem No 13

Try this beautiful Problem on Geometry from AMC 10A, 2019.Problem-13. You may use sequential hints to solve the problem.

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.