MathDB

Paul Revere is currently at

\left(x_0, y_0\right)

in the Cartesian plane, which is inside a triangle-shaped ship with vertices at

\left(-\dfrac{7}{25},\dfrac{24}{25}\right),\left(-\dfrac{4}{5},\dfrac{3}{5}\right)

, and

\left(\dfrac{4}{5},-\dfrac{3}{5}\right)

. Revere has a tea crate in his hands, and there is a second tea crate at

(0,0)

. He must walk to a point on the boundary of the ship to dump the tea, then walk back to pick up the tea crate at the origin. He notices he can take 3 distinct paths to walk the shortest possible distance. Find the ordered pair

(x_0, y_0)

.
Proposed by Derek Zhao
Solution.

\left(-\dfrac{7}{25},\dfrac{6}{25}\right)

Let

L

M

, and

N

be the midpoints of

BC

AC

, and

AB

, respectively. Let points

D

E

, and

F

be the reflections of

O = (0,0)

over

BC

AC

, and

AB

, respectively. Notice since

MN \parallel BC

BC \parallel EF

. Therefore,

O

is the orthocenter of

DEF

. Notice that

(KMN)

is the nine-point circle of

ABC

because it passes through the midpoints and also the nine-point circle of

DEF

because it passes through the midpoints of the segments connecting a vertex to the orthocenter. Since

O

is both the circumcenter of

ABC

and the orthocenter of

DEF

and the triangles are

180^\circ

rotations of each other, Revere is at the orthocenter of

ABC

. The answer results from adding the vectors

OA +OB +OC

, which gives the orthocenter of a triangle.

Problem Statement