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# Area of Equilateral Triangle | AIME I, 2015 | Question 4

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2015 from Geometry based on Area of Equilateral Triangle.

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2015 from Geometry, based on Area of Equilateral Triangle (Question 4).

## Area of Triangle – AIME I, 2015

Point B lies on line segment AC with AB =16 and BC =4. Points D and E lie on the same side of line AC forming equilateral triangle ABD and traingle BCE. Let M be the midpoint of AE, and N be the midpoint of CD. The area of triangle BMN is x. Find $x^{2}$.

• is 107
• is 507
• is 840
• cannot be determined from the given information

### Key Concepts

Algebra

Theory of Equations

Geometry

AIME, 2015, Question 4

Geometry Revisited by Coxeter

## Try with Hints

Let A(0,0), B(16,0),C(20,0). let D and E be in first quadrant. then D =$(8,8\sqrt3)$, E=$(18,2\sqrt3$).

M=$(9,\sqrt3)$, N=($14,4\sqrt3$), where M and N are midpoints

since BM, BN, MN are all distance, BM=BN=MN=$2\sqrt13$. Then, by area of equilateral triangle, x=$13\sqrt3$ then$x^{2}$=507.