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# Some old geometry problem for you to solve

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Sorry if it is hard to read.$$In triangle ABC, let the angle bisectors be \overline{BY} and \overline{CZ}. Given AB = 8, AY = 6, and CY = 3, find BC. [asy] pair A,B,C,X,Y,Z,I; A= (0,0); B = (1,0); C = (0.8,0.7); X = intersectionpoint(B--C , A -- (bisectorpoint(B,A,C))); Y = intersectionpoint(A--C, B -- scale(6)*( (bisectorpoint(C,B,A)) - B)); I = intersectionpoint(A--X, B--Y); Z = extension(A,B,C,I); draw(A--B--C--cycle); draw(B--Y); draw(C--Z); label("A",A,SW); label("B",B,SE); label("C",C,N); label("I",I,SW); label("Y",Y,NW); label("Z",Z,S); [/asy]$$

Feb 3, 2022