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2022 MMATHS Team Online p5 - min sum of 2 areas, starting with equilateral

Source:

October 1, 2023

geometryMMATHS

Problem Statement

Equilateral triangle

\vartriangle ABC

has side length

6

. Points

D

and

E

lie on

\overline{BC}

such that

BD = CE

and

B

,

D

,

E

,

C

are collinear in that order. Points

F

and

G

lie on

\vartriangle BFD

such that

\overline{FD} \perp \overline{BC}

, and

GF = GA

. If the minimum possible value of the sum of the areas of

\vartriangle BFD

and

gcd (a, c) = 1

can be expressed as

\frac{a\sqrt{b}}{c}

for positive integers

a, b, c

with

gcd (a, c) = 1

and

b

squarefree, find

a + b + c

.

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