<prompt> Pembetatu ABC ina urefu wa upande AB4, BC5, na CA6. Alama D na E ziko kwenye ray AB na AB  <response>: asy unitsize20; jozi A, B, C, D, E, F, X, O1, O2; A 0, 0; B 4, 0; C sehemu za makutanomduaraA, 6, duaraB,50; D B 54 1 sqrt 2, 0; E D 4 sqrt2, 0; F sehemu za makutanomduaraD, 2, duaraE, 71; X uganiA, E, C, F; O1 circumcenterC, A, D; O2 circumcenterC, B, E; filldrawA--B--C--mzunguko, lightcyan, deepcyan; filldrawD--E--F--mzunguko, lightmagenta, deepmagenta; kuchora B--D, kijivu 0.6; kuchora C--F, kijivu 0.6; chora mduara C, A, D, iliyopigwa; kuchora mduara C, B, E, iliyopigwa; nuktaA, A, dirA-O1; nuktaB, B, dir240; nuktaC, C, dir120; nuktaD, D, dir40; nuktaE, E, dirE-O2; nuktaF, F, dir270; nuktaX, X, dir140; lebo6, CA2, dirC-AI, deepcyan; lebo5, CB2, dirB-CI, deepcyan; lebo4, AB2, dirA-BI, deepcyan; lebo7, FE2, dirF-EI, deepmagenta; lebo2, FD2, dirD-FI, deepmagenta; lebo4sqrt2, DE2, dirE-DI, deepmagenta; leboa, BX2, dirB-XI, kijivu0.3; leboasqrt2, DX2, dirD-XI, kijivu0.3; asyOna kwambapembe DFEpembe CFE-pembe CFDpembe CBE-angle CAD180-B-AC.Kwa Sheria ya Cosines,cos CfracAC2 BC2-AB22cdot ACcdot BCfrac34.Kisha,DE2DF2EF2-2cdot DFcdot EFcos C 32inamaanisha DE4sqrt2.Ruhusu XoverlineABcapoverlineCF, aXB, na bXD. Kisha,XAcdot XDXCcdot XFXBcdot XEinamaanisha ba4ab4sqrt2inamaanisha basqrt2.Hata hivyo, kwa kuwa pembetatu XFDsimpembetatu XAC, XFtfrac4a3, lakini tangu pembetatu XFEsimpembetatu XBC,frac75frac4a3a inamaanisha afrac54inamaanisha BEaasqrt24sqrt2frac521sqrt24,na jumla iliyoombwa ni 5212432. 