2024 AIME I Problem 5

Question

Rectangle

ABCD

has dimensions

AB = 107

and

BC = 16,

and rectangle

EFGH

has dimensions

EF = 184

and

FG = 17.

Points

D,

E,

C,

and

F

lie on line

DF

in that order, and

A

and

H

lie on opposite sides of line

DF,

as shown. Points

A,

D,

H,

and

G

lie on a common circle. Find

CE.

Answer

Answer: 104

Concepts:coordinate geometry circle perpendicular bisector

Difficulty rating: 2390

Small hint:

Put line

DF

on the

x

-axis with

D

at the origin, and write the coordinates of

A,

D,

H,

and

G

in terms of

DE

Big hint:

The circle’s center lies on

y = -8

(bisector of

\overline{AD}

) and on the vertical bisector of

\overline{HG};

equate its distances to

D

and

H

Solution:

Put line

DF

on the

x

-axis with

D = (0, 0)

and

C = (107, 0),

so

A = (0, -16).

Let

DE = e.

Then

E = (e, 0),

F = (e + 184, 0),

and the second rectangle sits above the line:

H = (e, 17)

and

G = (e + 184, 17).

The center of the circle through

A,

D,

H,

G

lies on the perpendicular bisector of the vertical segment

\overline{AD},

the line

y = -8,

and on the perpendicular bisector of the horizontal segment

\overline{HG},

the line

x = e + 92.

Equating the center’s squared distances to

D

and to

H,

\begin{aligned} &(e + 92)^2 + 8^2 \\ &= 92^2 + 25^2 = 9089, \end{aligned}

so

(e + 92)^2 = 9025

and

e + 92 = 95,

giving

e = 3.

Therefore

CE = DC - DE

= 107 - 3 = 104.