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December 7, 2015

West Bengal RMO 2015 Problem 5 Solution - Equilateral Triangle


The second stage examination of INMO, the Regional Mathematical Olympiad (RMO) is a three hour examination with six problems. The problems under each topic involve high level of difficulty and sophistication. The book, Challenge and Thrill of Pre-College Mathematics is very useful for preparation of RMO. West Bengal RMO 2015 Problem 5 Solution has been written for RMO preparation series.


Problem:


Let ABC be a triangle with circumcircle \Gamma and incenter I. Let the internal angle bisectors of \angle A,\angle B,\angle C meet \Gamma in A',B',C' respectively. Let B'C' intersect AA' at P, and AC in Q. Let BB' intersect AC in R. Suppose the quadrilateral PIRQ is a kite; that is, IP=IR and QP=QR. Prove that ABC is an equilateral triangle.


Discussion:


Screen Shot 2015-12-06 at 10.41.28 PM

Since PIRQ is a kite, \Delta PIQ \equiv \Delta RIQ  by side-side-side congruence rule. Hence \angle IRQ (or \angle IRA) = \angle IPQ (or \angle IPB')

Also \angle AIR = \angle B'IP  (same angle).

Hence \Delta AIR, \Delta B'IP  are equiangular. This implies \angle IAR = \angle IB'P

But \angle IB'P = \angle BCC'  (because both of them are subtended by the same segment BC').

Hence \angle BCC' \left (=\dfrac{\angle C}{2}\right ) = \angle IAR \left (=\dfrac {\angle A}{2} \right)  implying \angle A = \angle C

Screen Shot 2015-12-06 at 11.18.07 PM

Since ABC is isosceles (BA = BC) and BR is the angle bisector, hence it is perpendicular to AC. Thus \angle BRA = \angle IPQ = 90^o  .

Join AB'. \angle AB'P = \angle ACC' = \dfrac{\angle C}{2}  since they are subtended by the same segment AC' and CC' is the angle bisector of C.
Similarly \angle IB'P = \angle BCC' = \dfrac{\angle C}{2}  since they are subtended by the same segment BC' and CC' is the angle bisector of C.

Clearly \angle IB'P = \angle AB'P = \dfrac{\angle C}{2}  .
Since earlier we found \angle IPB' = \angle APB' = 90^o  ,
therefore \Delta AB'P  is isosceles with AB'=IB'.

Finally AI = IB' (because \Delta AIR \equiv \Delta B'IP  since earlier we proved them to be equiangular, and we can also show that AR = B'P; How? Clearly \Delta AQP \equiv \Delta B'QR  as QP= QR, two right angles equal, vertically opposite angles equal. Thus AQ = QB'. QR = QP'. Adding these two we get AR = PB').

Since AI = IB' and earlier we showed IB' = AB' hence the triangle \Delta AIB'  is equilateral, implying \angle AB'I = \angle ACB = 60^o . Since ABC is an isosceles triangle with one angle 60^o  hence it is equilateral.


Chatuspathi:

  • What is this topic: Geometry
  • What are some of the associated concept: Cyclic Quadrilateral
  • Where can learn these topics:  Cheenta Math Olympiad Program, discuss these topics in the ‘Geometry’ module.
  • Book Suggestions: Challenges and Thrills of Pre-College Mathematics

7 comments on “West Bengal RMO 2015 Problem 5 Solution - Equilateral Triangle”

  1. but i could solve it without the information on kite just as i is the incentre so point R is the point where the incircle meets and its right angle similarly on the left hand side also and just consider the two right triangles RIC and just vertically opposite to it and provet the rest same way......

    1. R is not necessarily the point where incircle meets. It is the point where angle bisector meets the opposite side. You must drop perpendicular from I on the sides to get the points through which incircle pass

  2. Beautiful proof.. but there's just one type error.. ∆AB'P is not isoceles.. it will be ∆AB'I instead..

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