2015 USAJMO Problems/Problem 3
Contents
[hide]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
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