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# INMO 2013 Question No. 1 Solution 1     Let and be two circles touching each other externally at R. Let and be the centres of and , respectively. Let be a line which is tangent to at P and passing through , and let be the line tangent to at Q and passing through . Let . If KP=KQ then prove that the triangle PQR is equilateral.

Discussion:

We note that is a straight line (why?)
Also are right angled triangles with right angles at point Q and P respectively.
Hence are similar (vertically opposite angles and right angles)
Thus as KP =KQ.
Hence the radii of the two circles are equal..
This implies R is the midpoint of hence the midpoint of hypotenuse of  since all are circum-radii of .
Hence is equilateral, similarly is also equilateral.
Thus is also .
Hence triangle PQR is equilateral.

1     Let and be two circles touching each other externally at R. Let and be the centres of and , respectively. Let be a line which is tangent to at P and passing through , and let be the line tangent to at Q and passing through . Let . If KP=KQ then prove that the triangle PQR is equilateral.

Discussion:

We note that is a straight line (why?)
Also are right angled triangles with right angles at point Q and P respectively.
Hence are similar (vertically opposite angles and right angles)
Thus as KP =KQ.
Hence the radii of the two circles are equal..
This implies R is the midpoint of hence the midpoint of hypotenuse of  since all are circum-radii of .
Hence is equilateral, similarly is also equilateral.
Thus is also .
Hence triangle PQR is equilateral.

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### One comment on “INMO 2013 Question No. 1 Solution”

1. Anonymous says:

the solution to first question isn't complete...you will also hav to contradict the case where tangents are opposite

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