120 angle

Geometry Level 4

Let A B C ABC be a triangle with B A C = 12 0 \angle BAC=120^{\circ} . Suppose that the bisectors of angles B A C , A B C \angle BAC, \angle ABC and A C B \angle ACB meet sides B C , A C BC, AC and A B AB at points D , E D, E and F F , respectively.

Find the measure of the largest angle of triangle D E F DEF (in degrees).


The answer is 90.

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1 solution

Ajit Athle
Jan 28, 2018

Use the nomenclature as shown. Extend BA to say 'X'. <XAC=60° since <BAC=120° and this implies that AC bisects <XAE Now consider the triangle ABE in which we have one internal angle bisector and one external angle bisector intersecting in point F. Hence we conclude that EF bisects the <AEC externally. Likewise, ED bisects <AEB externally. Since, ED & EF are bisectors of <AEB & <AEC respectively, ED is perpendicular to EF or <DEF=90°.

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