<prompt> Miduara omega_1, omega_2, na omega_3 kila moja ina kipenyo cha 4 na imewekwa kwenye ndege ili kila duara iwe na mduara wa nje kwa mbili nyingine. Alama P_1, P_2, na P_3 zinapatikana kwenye omega_1, omega_2, na omega_3 mtawalia kiasi kwamba P_1P_2P_2P_3P_3P_1 na laini P_iP_i1 ni tangent hadi omega_i kwa kila i1,2,3, ambapo P_4 P_1. Tazama takwimu hapa chini. Eneo la pembetatu P_1P_2P_3 linaweza kuandikwa kwa njia ya sqrtasqrtb kwa nambari kamili a na b. ab ni nini?asy unitsize12; jozi A 0, 8sqrt3, B zungusha-120A, C zungusha120A; theta halisi 41.5; jozi P1 zungushatheta22sqrt73, 0, P2 zungusha-120P1, P3 zungusha120P1; kujaza P1--P2--P3--mzunguko, kijivu 0.9; choraMduaraA, 4; chora MduaraB, 4; chora MduaraC, 4; nukta P1; nukta P2; nukta P3; defaultpenfontsize10pt; leboP_1, P1, E1.5; leboP_2, P2, SW1.5; leboP_3, P3, N; leboomega_1, A, W17; leboomega_2, B, E17; leboomega_3, C, W17; asytextbfA 546qquadtextbfB 548qquadtextbfC 550qquadtextbfD 552qquadtextbfE 554  <response>: Acha O_i iwe kitovu cha mduara omega_i kwa i1,2,3, na K iwe makutano ya mistari O_1P_1 na O_2P_2. Kwa sababu angle P_1P_2P_3 60circ, inafuata kwamba pembetatu P_2KP_1 ni 30-60-90circ pembetatu. Acha dP_1K; kisha P_2K 2d na P_1P_2 sqrt 3d. Sheria ya Cosines katika pembetatu O_1KO_2 inatoa82 d42 2d-42 - 2d42d-4cos 60circ ,ambayo hurahisisha hadi 3d2 - 12d - 16 0. Suluhisho chanya ni d 2 tfrac23sqrt21. Kisha P_1P_2 sqrt 3 d 2sqrt 3 2sqrt 7, na eneo linalohitajika nifracsqrt 34cdotleft2sqrt 3 2sqrt 7. kulia2 10sqrt 3 6sqrt 7 sqrt300 sqrt252.Jumla iliyoombwa ni 300 252 552. 