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.

# Problem

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

• 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.

Also see

## 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,

# Reference:

• 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.