Difference between revisions of "2015 USAMO Problems/Problem 2"
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Next, consider the homothety of scale factor <math>\frac{2}{3}</math> about <math>O</math> mapping <math>N_9</math> to <math>G</math>. This means that the locus of <math>G</math> is a circle as well. | Next, consider the homothety of scale factor <math>\frac{2}{3}</math> about <math>O</math> mapping <math>N_9</math> to <math>G</math>. This means that the locus of <math>G</math> is a circle as well. | ||
− | Finally, we take a homothety of scale factor <math>\frac{3}{2}</math> about <math>A</math> mapping <math>G</math> to <math>M</math>. Hence, the locus of <math>M</math> is a circle, as desired. | + | Finally, we take a homothety of scale factor <math>\frac{3}{2}</math> about <math>A</math> mapping <math>G</math> to <math>M</math>. Hence, the locus of <math>M</math> is a circle, as desired. - Spacesam |
===Fake Solution=== | ===Fake Solution=== | ||
Note that each point <math>X</math> on <math>PQ</math> corresponds to exactly one point on arc <math>PBQ</math>. Also notice that since <math>AB</math> is the diameter of <math>\omega</math>, <math>\angle ASB</math> is always a right angle; therefore, point <math>T</math> is always <math>B</math>. WLOG, assume that <math>\omega</math> is on the coordinate plane, and <math>B</math> corresponds to the origin. The locus of <math>M</math>, since the locus of <math>S</math> is arc <math>PBQ</math>, is the arc that is produced when arc <math>PBQ</math> is dilated by <math>\frac {1} {2}</math> with respect to the origin, which resides on the circle <math>\psi</math>, which is produced when <math>\omega</math> is dilated by <math>\frac {1} {2}</math> with respect to the origin. By MSmathlete1018 | Note that each point <math>X</math> on <math>PQ</math> corresponds to exactly one point on arc <math>PBQ</math>. Also notice that since <math>AB</math> is the diameter of <math>\omega</math>, <math>\angle ASB</math> is always a right angle; therefore, point <math>T</math> is always <math>B</math>. WLOG, assume that <math>\omega</math> is on the coordinate plane, and <math>B</math> corresponds to the origin. The locus of <math>M</math>, since the locus of <math>S</math> is arc <math>PBQ</math>, is the arc that is produced when arc <math>PBQ</math> is dilated by <math>\frac {1} {2}</math> with respect to the origin, which resides on the circle <math>\psi</math>, which is produced when <math>\omega</math> is dilated by <math>\frac {1} {2}</math> with respect to the origin. By MSmathlete1018 |
Revision as of 17:38, 28 July 2019
Problem
Quadrilateral is inscribed in circle with and . Let be a variable point on segment . Line meets again at (other than ). Point lies on arc of such that is perpendicular to . Let denote the midpoint of chord . As varies on segment , show that moves along a circle.
Solution 1
We will use coordinate geometry.
Without loss of generality, let the circle be the unit circle centered at the origin, , where .
Let angle , which is an acute angle, , then .
Angle , . Let , then .
The condition yields: (E1)
Use identities , , , we obtain . (E1')
The condition that is on the circle yields , namely . (E2)
is the mid-point on the hypotenuse of triangle , hence , yielding . (E3)
Expand (E3), using (E2) to replace with , and using (E1') to replace with , and we obtain , namely , which is a circle centered at with radius .
Solution 2
Let the midpoint of be . We claim that moves along a circle with radius .
We will show that , which implies that , and as is fixed, this implies the claim.
by the median formula on .
by the median formula on .
.
As , from right triangle .
By , .
Since is the circumcenter of , and is the circumradius, the expression is the power of point with respect to . However, as is also the power of point with respect to , this implies that .
By ,
Finally, by AA similarity ( and ), so .
By , , so , as desired.
Solution 3(synthetic)
To begin with, we connect and we construct the nine-point circle of centered at . Lemma : We proceed on a directed angle chase. We get , so and the desired result follows by side length ratios.
Lemma : The locus of as moves along is a circle centered about . We add the midpoint of , , and let the circumradius of be . Taking the power of with respect to , we get Hence, , which remains constant as moves.
Next, consider the homothety of scale factor about mapping to . This means that the locus of is a circle as well.
Finally, we take a homothety of scale factor about mapping to . Hence, the locus of is a circle, as desired. - Spacesam
Fake Solution
Note that each point on corresponds to exactly one point on arc . Also notice that since is the diameter of , is always a right angle; therefore, point is always . WLOG, assume that is on the coordinate plane, and corresponds to the origin. The locus of , since the locus of is arc , is the arc that is produced when arc is dilated by with respect to the origin, which resides on the circle , which is produced when is dilated by with respect to the origin. By MSmathlete1018