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Print2003 Vietnamese Mathematical Olympiad
Vietnam 2003 geometry
Problem
In plane, let be given two fixed circles and touching each other at , the radius of is greater than that of . is the point on such that the points , , are not collinear. Let and be the tangents to with touching points and . The lines and cut again respectively at and . Let be the point of intersection of the line and tangent to at . Prove that moves on a fixed line when moves on so that the three points , , are not collinear.
Solution
We consider two cases:
First case: The circles touch each other externally at . Let be the common tangent at of . Since and are tangents to at and , we have . But , hence . As it follows that . The triangles and have: and , so they are similar. Therefore t.c. . But (the power of with respect to the circle ) , where is the radius of , it follows that . Analogously, we have . So As lies on the line , it implies that , i.e.
Since is the tangent to at , (2), where is the radius of . From (1) and (2), we get , or . It shows that lies on the radical line of and .
2d case: The circles touch each other internally at . The proof in this case is analogous to the proof in the 1st case.
First case: The circles touch each other externally at . Let be the common tangent at of . Since and are tangents to at and , we have . But , hence . As it follows that . The triangles and have: and , so they are similar. Therefore t.c. . But (the power of with respect to the circle ) , where is the radius of , it follows that . Analogously, we have . So As lies on the line , it implies that , i.e.
Since is the tangent to at , (2), where is the radius of . From (1) and (2), we get , or . It shows that lies on the radical line of and .
2d case: The circles touch each other internally at . The proof in this case is analogous to the proof in the 1st case.
Techniques
TangentsRadical axis theoremAngle chasingConstructions and loci