Difference between revisions of "1991 USAMO Problems/Problem 5"
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We note that | We note that | ||
− | <cmath> CE = CC_a - EC_a = CC_b - | + | <cmath> CE = CC_a - EC_a = CC_b - EC_b = \frac{CC_a + CC_b - (EC_a + EC_b)}{2} . </cmath> |
On the other hand, since <math>EC_a</math> and <math>ET_a</math> are tangents from the same point to a common circle, <math>EC_a = T_aE</math>, and similarly <math>EC_b = ET_b</math>, so | On the other hand, since <math>EC_a</math> and <math>ET_a</math> are tangents from the same point to a common circle, <math>EC_a = T_aE</math>, and similarly <math>EC_b = ET_b</math>, so | ||
<cmath> EC_a + EC_b = T_aE + ET_b = T_a T_b . </cmath> | <cmath> EC_a + EC_b = T_aE + ET_b = T_a T_b . </cmath> |
Revision as of 00:51, 19 August 2012
Problem
Let be an arbitrary point on side
of a given triangle
and let
be the interior point where
intersects the external common tangent to the incircles of triangles
and
. As
assumes all positions between
and
, prove that the point
traces the arc of a circle.
Solution
Let the incircle of and the incircle of
touch line
at points
, respectively; let these circles touch
at
,
, respectively; and let them touch their common external tangent containing
at
, respectively, as shown in the diagram below.
We note that
On the other hand, since
and
are tangents from the same point to a common circle,
, and similarly
, so
On the other hand, the segments
and
evidently have the same length, and
, so
. Thus
If we let
be the semiperimeter of triangle
, then
, and
, so
Similarly,
so that
Thus
lies on the arc of the circle with center
and radius
intercepted by segments
and
. If we choose an arbitrary point
on this arc and let
be the intersection of lines
and
, then
becomes point
in the diagram, so every point on this arc is in the locus of
.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
Resources
1991 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |