INMO-Problem-2!

Geometry Level 3

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

Submit your answer to two decimal places.

The answer is 20.78.

1 solution

Anish Kulkarni
Jan 18, 2019

ΔAFC has AE = EC and NF = NC = 3 therefore by MPT EN||AF so ED||BA, but E is midpoint of AC therefore by converse of MPT D is midpoint from here on u can use simple non-construction angle chasing and simple properties of right angled Δ to get ΔABC is equilateral, then ΔADC is 30-60-90 Δ and MC is angle bisector of 60 and we know its length to be 3 + 1 = 4 so then we calculate side length and thus perimeter. Depending on what value of √3 u take u may get a value slightly off from the value given as solution, but as long as u have figured this part out and have gotten an ans in this range it should be fine

INMO-Problem-2!

The answer is 20.78.

1 solution

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