IMO Shortlist 1982 problem 13
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Avg: 0,0 A non-isosceles triangle
has sides
,
,
with the side
lying opposite to the vertex
. Let
be the midpoint of the side
, and let
be the point where the inscribed circle of triangle
touches the side
. Denote by
the reflection of the point
in the interior angle bisector of the angle
. Prove that the lines
,
and
are concurrent.
![A_{1}A_{2}A_{3}](/media/m/e/7/3/e73ef0f47bf796f8f833006f4c93890e.png)
![a_{1}](/media/m/0/6/5/0653090dabb5d1972cd7a7dfcd31abc1.png)
![a_{2}](/media/m/5/5/6/5565dac5c7f1dadb0e60c273c1d11c80.png)
![a_{3}](/media/m/4/b/1/4b1295ff3d20438f431fda51aeded085.png)
![a_{i}](/media/m/3/9/8/39813f04a4f0ae199804be0f84564cfe.png)
![A_{i}](/media/m/6/1/a/61ae488afb5102f36f7e70390d6c0d8a.png)
![M_{i}](/media/m/2/c/4/2c40290e5d0c9c51c525a626689c3796.png)
![a_{i}](/media/m/3/9/8/39813f04a4f0ae199804be0f84564cfe.png)
![T_{i}](/media/m/4/b/0/4b0374597bca7311b66530bc20096227.png)
![A_{1}A_{2}A_{3}](/media/m/e/7/3/e73ef0f47bf796f8f833006f4c93890e.png)
![a_{i}](/media/m/3/9/8/39813f04a4f0ae199804be0f84564cfe.png)
![S_{i}](/media/m/9/8/c/98c9204010a13b731c4185111db5091d.png)
![T_{i}](/media/m/4/b/0/4b0374597bca7311b66530bc20096227.png)
![A_{i}](/media/m/6/1/a/61ae488afb5102f36f7e70390d6c0d8a.png)
![M_{1}S_{1}](/media/m/8/a/c/8ac91cc8d19719ce98b27e7c11ab615b.png)
![M_{2}S_{2}](/media/m/b/8/2/b823b8fbebc93d3ec84c42d145df7ea7.png)
![M_{3}S_{3}](/media/m/e/3/2/e32eae45d2012f7332f5626eec0ba1fe.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1982