On the construction of a triangle from the feet of its angle bisectors

On the construction of a triangle from the feet of its angle bisectors Alexey V. Ustinov ∗ No v em ber 12, 2018 Giv en a tr iangle AB C with sides a , b , c . W e wan t to construct a tr iangle A ′ B ′ C ′ suc h that segmen ts AA ′ , B B ′ and C C ′ are its angle bisectors. This problem was discussed in [3, 2] (see problem 1 38) and remained op en. Ar ticle [4] giv es conic solution of this problem. Using form ulae from [4] it is easy to pro v e that in the general case construction of a triangle from the feet of its angle bisectors with compass and ruler is imp ossible. F ollo wing this a rticle w e denote by ( x : y : z ) b arycen tric co ordinates o f incen te r of triangle A ′ B ′ C ′ with resp ect to triangle AB C . Th en vertice s of triangle A ′ B ′ C ′ will ha v e co ordinates ( − x, y , z ), ( x, − y , z ), ( x, y , − z ). As it w as pr o v ed in [4], n um b ers x , y , z satisfy follo wing equations: − x ( c 2 y 2 − b 2 z 2 ) + y z (( c 2 + a 2 − b 2 ) y − ( a 2 + b 2 − c 2 ) z ) =0 , − y ( a 2 z 2 − c 2 x 2 ) + xz (( a 2 + b 2 − c 2 ) z − ( b 2 + c 2 − a 2 ) x ) =0 , − z ( b 2 x 2 − a 2 y 2 ) + xy (( b 2 + c 2 − a 2 ) x − ( c 2 + a 2 − b 2 ) y ) =0 . Third equation follo ws from first and second b ecause in our situation xy z 6 = 0. Let a = 2, b = 3, c = √ 7 ( ∠ C = 60 ◦ ). Using fi rst equation we can w r ite x in terms of y and z . Substitutions into the second equation lead to the irr educible equation for t = z / (3 y ): t 3 + 74 t 2 + 259 t − 570 = 0 . It has three ro ots t 1 = 1 . 52 . . . , t 2 = − 5 . 32 . . . , t 3 = − 70 . 19 . . . . Firs t one giv es the incen ter of some triangle A ′ B ′ C ′ , another ones co rresp ond to excente rs (of anot her triangle s). It is imp ossible to construct t 1 using compass and ruler (see [1, c h. 3]), hence w e can’t construct the incen ter of triangle A ′ B ′ C ′ and its v ertices. Author is greatful for V. Dub ro vsky for the references [2, 3]. References [1] Courant R., R obbins H. W hat is mathemat ics? Oxford Universit y Press, 1979. [2] Meyers L. F. Up date on William W ernic k’s “T r iangle Con s tructions with Thr ee L o cated P oin ts” — Math. Mag. , 69 (1996), 46–49. [3] Wernick W. T riangle Constructions with Thr ee Lo cated Poin ts. — Math. Mag. , 55 (1982), 227–2 30. [4] Yiu P. Conic construction of a triangle from th e feet of its angle bisectors. — J. Ge om. Gr aph. , 12 (2008 ), 171–182 . ∗ The researc h of the author w as supported by Dynasty F oundation 1