Difference between revisions of "2021 AIME I Problems/Problem 13"
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− | Let <math>O_i</math> and <math>r_i</math> be the center and radius <math>\omega_i</math>, and let <math>O</math> and <math>r</math> be the center and radius of <math>\omega</math>. | + | Let <math>O_i</math> and <math>r_i</math> be the center and radius of <math>\omega_i</math>, and let <math>O</math> and <math>r</math> be the center and radius of <math>\omega</math>. |
Since <math>\overline{AB}</math> extends to an arc with arc <math>120^\circ</math>, the distance from <math>O</math> to <math>\overline{AB}</math> is <math>r/2</math>. Let <math>X=\overline{AB}\cap \overline{O_1O_2}</math>. Consider <math>\triangle OO_1O_2</math>. The line <math>\overline{AB}</math> is perpendicular to <math>\overline{O_1O_2}</math> and passes through <math>X</math>. Let <math>H</math> be the foot from <math>O</math> to <math>\overline{O_1O_2}</math>; so <math>HX=r/2</math>. We have by tangency <math>OO_1=r+r_1</math> and <math>OO_2=r+r_2</math>. Let <math>O_1O_2=d</math>. | Since <math>\overline{AB}</math> extends to an arc with arc <math>120^\circ</math>, the distance from <math>O</math> to <math>\overline{AB}</math> is <math>r/2</math>. Let <math>X=\overline{AB}\cap \overline{O_1O_2}</math>. Consider <math>\triangle OO_1O_2</math>. The line <math>\overline{AB}</math> is perpendicular to <math>\overline{O_1O_2}</math> and passes through <math>X</math>. Let <math>H</math> be the foot from <math>O</math> to <math>\overline{O_1O_2}</math>; so <math>HX=r/2</math>. We have by tangency <math>OO_1=r+r_1</math> and <math>OO_2=r+r_2</math>. Let <math>O_1O_2=d</math>. |
Revision as of 23:40, 13 March 2021
Problem
Circles and
with radii
and
, respectively, intersect at distinct points
and
. A third circle
is externally tangent to both
and
. Suppose line
intersects
at two points
and
such that the measure of minor arc
is
. Find the distance between the centers of
and
.
Solution
Let and
be the center and radius of
, and let
and
be the center and radius of
.
Since extends to an arc with arc
, the distance from
to
is
. Let
. Consider
. The line
is perpendicular to
and passes through
. Let
be the foot from
to
; so
. We have by tangency
and
. Let
.
Since
is on the radical axis of
and
, it has equal power, so
since
. Now we can solve for
and
, and in particular,
We want to solve for
. By the Pythagorean Theorem (twice):
Therefore,
.
See also
2021 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.