Incentre midpoint triangle

Geometry Level 2

Let A B C ABC be a triangle with incentre I I . Let M , N M, N be the midpoints of A B , A C AB, AC respectively. Let O O be the circumcentre of triangle M N I MNI . Find the first two decimal places of the length of A O AO if A B = 13 , B C = 14 , C A = 15 AB = 13, BC = 14, CA = 15 .


The answer is 6.

Adhavan Sv by Adhavan SV , 17, India

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1 solution

Adhavan Sv
Mar 31, 2019

Let D, E be the foot of perpendiculars from I to AB, AC. Note that MD = NE = 0.5. Also, ID = IE. So, triangle IDM and triangle IDN are congruent. So, this gives us that A, M, N, I are concyclic. But we also know that A, M, N, X are concyclic where X is the circumcentre of triangle ABC. So, the circumcentre O of triangle MNI is the midpoint of AX. Now the rest is just plain computations.

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