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Question

Let ABC be an acute angled triangle in which D, E, F are points on BC, CA, AB respectively such that AD BC, AE=EC and CF bisects C internally. Suppose CF meets AB and DE in M and N respectively. If FM=2, MN=1, NC=3, Find the perimeter of the triangle ABC.

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Solution

¯¯¯¯¯¯¯¯¯FN=¯¯¯¯¯¯¯¯¯NC=3 and ¯¯¯¯¯¯¯¯AE=¯¯¯¯¯¯¯¯EC (given)
So, According to SAS Similarity,
NECFAC

So, ¯¯¯¯¯¯¯¯¯NE||¯¯¯¯¯¯¯¯AF¯¯¯¯¯¯¯¯AB||¯¯¯¯¯¯¯¯¯ED

ie, ¯¯¯¯¯¯¯¯¯AD is a perpendicular bisector
ABC is an isosceles triangle.

Also, ¯¯¯¯¯¯¯¯FC passes through centroid because ¯¯¯¯¯¯¯¯¯¯FM:¯¯¯¯¯¯¯¯¯¯MC=2:1

So, M is the centroid and incentre of the triangle.
ie, ¯¯¯¯¯¯¯¯¯¯FM=¯¯¯¯¯¯¯¯¯¯MD=2=r (inradius)

And ¯¯¯¯¯¯¯¯¯AD=6
¯¯¯¯¯¯¯¯¯DC=¯¯¯¯¯¯¯¯¯¯MC2¯¯¯¯¯¯¯¯¯¯MD2=4222=23=12¯¯¯¯¯¯¯¯BC

Area=12¯¯¯¯¯¯¯¯¯ADׯ¯¯¯¯¯¯¯BC=6×23=12r(Perimeter)

So, Perimeter=123

787614_696748_ans_d818b2b486c44836b45f76ccc678d958.png

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