RMO 2018 Tamil Nadu Problem 1 is from Geometry. We present sequential hints for this problem. Do not read all hints at one go. Try it yourself.
Let ABC be an acute-angled triangle and let D be an interior point of the line segment BC. Let the circumcircle of triangle ACD intersect AB at E (E between A and B) and let the circumcircle of triangle ABD intersect AC at F (F between A and C). Let O be the circumcenter triangle AEF. Prove that OD bisects \( \angle EDF \)
Key ideas you will need to solve this problem
- Properties of Cyclic Quadrilaterals:
- Opposite angles of a cyclic quadrilateral adds up to \( 180^o \)
- Angle at the center is twice the angle at the circumference.
- Angles subtended by the same arc at circumference are equal.
Hint 1: Draw a picture
Hint 2: A construction would be nice
Join OE and OF. What kind of quadrilateral is OEDF? (Pause here. Try it yourself.)
Hint 3: Some angle chasing
Note that \( EDCA \) is a cyclic quadrilateral (all of its four vertices are on a circle). Hence \( \angle ODB = \angle EAC = \angle A \) .
Similarly \( AFDB \) is cyclic. Hence \(\angle CDF = \angle FAB = \angle A \)
This implies \( \angle EDF = 180^o – 2 \angle A \). Since O is the center of \( \Delta AEF \), \( \angle EOF = 2\times \angle EAF = 2 \times \angle A\).
Therefore \( \angle EDF + \angle EAF = 180^o – 2 \times \angle A + 2 \times \angle A = 180^o \). This implies quadrilateral OEDF is cyclic.
Hint 4: OE = OF
OE = OF because O is the center and E, F are at the circumference of the circle passing through AEF. Therefore both are radii of the same circle.
This implies \( \angle OEF = \angle OFE \)
Hint 5: Final Lap; Some more angle chasing
Since OEDF is cyclic, hence \( \angle OEF = \angle ODF \) (angle subtended by the arc OF at the circumference).
Similarly \( \angle OFE = \angle ODE \).
But \( \angle OEF = \angle OFE \) (see hint 4).
Hence \( \angle ODE = \angle ODF \). The proof is complete,
- These ideas are usually discussed in the Geometry I module of Cheenta Math Olympiad Program.
- Challenges and Thrills of Pre -College Mathematics by Venkatchala is a good reference for some these ideas.