MathDB

BMT 2021 Geometry #10

Source:

August 12, 2023

geometry

Problem Statement

Consider

\vartriangle ABC

such that

CA + AB = 3BC

. Let the incircle

\omega

touch segments

\overline{CA}

and

\overline{AB}

at

E

and

F

, respectively, and define

P

and

Q

such that segments

\overline{P E}

and

\overline{QF}

are diameters of

\omega

. Define the function

D

of a point

K

to be the sum of the distances from

K

to

P

and

Q

(i.e.

D(K) = KP + KQ

). Let

W, X, Y

, and

Z

be points chosen on lines

\overleftrightarrow {BC}

,

\overleftrightarrow {CE}

,

\overleftrightarrow {EF}

, and

\overleftrightarrow {F B}

, respectively. Given that

BC =\sqrt{133}

and the inradius of

\vartriangle ABC

is

\sqrt{14}

, compute the minimum value of

D(W) + D(X) + D(Y ) + D(Z)

.

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