A non-isosceles triangle A_{1}A_{2}A_{3} has sides a_{1}, a_{2}, a_{3} with a_{i} opposite A_{i}. M_{i} is the midpoint of side a_{i} and T_{i} is the point where the incircle touches side a_{i}. Denote by S_{i} the reflection of T_{i} in the interior bisector of ∠A_{i}. Prove that the lines M_{1}S_{1}, M_{2}S_{2} and M_{3}S_{3} are concurrent.

**Solution**

Let B_{i} be the point of intersection of the interior angle bisector of the angle at A_{i} with the opposite side. The first step is to figure out which side of B_{i } T_{i} lies. Let A_{1} be the largest angle, followed by A_{2}. Then T_{2} lies between A_{1} and B_{2}, T_{3} lies between A_{1} and B_{3}, and T_{1} lies between A_{2} and B_{1}. For ∠OB_{2}A_{1} = 180^{o} - A_{1} - A_{2}/2 = A_{3} + A_{2}/2. But A_{3} + A_{2}/2 < A_{1} + A_{2}/2 and their sum is 180^{o}, so A_{3} + A_{2}/2 < 90^{o}. Hence T_{2} lies between A_{1} and B_{2}. Similarly for the others.

Let O be the center of the incircle. Then ∠T_{1}OS_{2} = ∠T_{1}OT_{2} - 2 ∠T_{2}OB_{2} = 180^{o} - A_{3} - 2(90^{o} - ∠OB_{2}T_{2}) = 2(A_{3} + A_{2}/2) - A_{3} = A_{2} + A_{3}. A similar argument shows ∠T_{1}OS_{3} = A_{2} + A_{3}. Hence S_{2}S_{3} is parallel to A_{2}A_{3}.

Now ∠T_{3}OS_{2} = 360^{o} - ∠T_{3}OT_{1} - ∠T_{1}OS_{2} = 360^{o} - (180^{o} - A_{2}) - (A_{2} + A_{3}) = 180^{o} - A_{3} = A_{1} + A_{2}. ∠T_{3}OS_{1} = ∠T_{3}OT_{1} + 2 ∠T_{1}OB_{1} = (180^{o} - A_{2}) + 2(90^{o} - ∠OB_{1}T_{1}) = 360^{o} - A_{2} - 2(A_{3} + A_{1}/2) = 2(A_{1} + A_{2} + A_{3}) - A_{2} - 2A_{3} - A_{1} = A_{1} + A_{2} = ∠T_{3}OS_{2}. So S_{1}S_{2} is parallel to A_{1}A_{2}. Similarly we can show that S_{1}S_{3} is parallel to A_{1}A_{3}.

So S_{1}S_{2}S_{3} is similar to A_{1}A_{2}A_{3} and turned through 180^{o}. But M_{1}M_{2}M_{3} is also similar to A_{1}A_{2}A_{3} and turned through 180^{o}. So S_{1}S_{2}S_{3} and M_{1}M_{2}M_{3} are similar and similarly oriented. Hence the lines through corresponding vertices are concurrent.

Solutions are also available in Murray S Klamkin, International Mathematical Olympiads 1978-1985, MAA 1986, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.

© John Scholes

jscholes@kalva.demon.co.uk

14 Oct 1998