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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,△NEC∼△FACSo, ¯¯¯¯¯¯¯¯¯NE||¯¯¯¯¯¯¯¯AF⇒¯¯¯¯¯¯¯¯AB||¯¯¯¯¯¯¯¯¯EDie, ¯¯¯¯¯¯¯¯¯AD is a perpendicular bisector⇒△ABC is an isosceles triangle.Also, ¯¯¯¯¯¯¯¯FC passes through centroid because ¯¯¯¯¯¯¯¯¯¯FM:¯¯¯¯¯¯¯¯¯¯MC=2:1So, M is the centroid and incentre of the triangle.ie, ¯¯¯¯¯¯¯¯¯¯FM=¯¯¯¯¯¯¯¯¯¯MD=2=r (inradius)And ¯¯¯¯¯¯¯¯¯AD=6¯¯¯¯¯¯¯¯¯DC=√¯¯¯¯¯¯¯¯¯¯MC2−¯¯¯¯¯¯¯¯¯¯MD2=√42−22=2√3=12¯¯¯¯¯¯¯¯BCArea=12¯¯¯¯¯¯¯¯¯AD×¯¯¯¯¯¯¯¯BC=6×2√3=12r(Perimeter)So, Perimeter=12√3

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