<prompt> Acha A,B,C, na D ziwe pointi kwenye mduara. Kwa kila jozi ya pointi, tunatoa mstari unaopita katikati ya pointi hizo mbili ambazo ni perpendicular kwa mstari unaoundwa na pointi nyingine mbili. Kwa mfano, tunachora mstari unaopita katikati ya overlineAB ambayo ni sawa na mstari wa CD. Kwa njia hii, tunachora jumla ya mistari binom42 6. .asyukubwa wa kitengo 2.5 cm;jozi A, B, C, D, P;jozi M;A dir118;B dir62;C dir323;D dir165;M12 A B2;M13 A C2;M14 A D2;M23 B C2;M24 B D2;M34 C D2;P A B C D2;choraMduara0,0,1,nyekundu;choraA--B,kijani;kuchoraA--C,kijani;choraA--D,kijani;kuchoraB--C,kijani;choraB--D,kijani;kuchoraC--D,kijani;kuchorainterpM12,P,-2--interpM12,P,6,iliyokatika;kuchorainterpM13,P,-2--interpM13,P,2,iliyokatika;kuchorainterpM14,P,-0.5--interpM14,P,2.5,iliyokatika;kuchorainterpM23,P,-0.7--interpM23,P,1.5,iliyokatika;kuchorainterpM24,P,-3.3--interpM24,P,4,iliyokatika;kuchorainterpM34,P,-0.3--interpM34,P,1.6, iliyokatika;nukta A, A, A;nukta B, B, B;nukta C, C, C;nukta D, D, D;nuktaM12;nukta M13;nuktaM14;nukta M23;nuktaM24;nukta M34;asyInabadilika kuwa mistari yote sita itapitia hatua sawa kila wakati, sema P. Kuna viambajengo a,b,c, na d hivi kwamba overrightarrowP inaweza kila wakati. kuonyeshwa kwa fomuarrow overrightarrowP overrightarrowA b overrightarrowB c overrightarrowC d overrightarrowD.Tafuta a b c d.  <response>: Fikiria mstari unaopita katikati ya overlineAB. Mstari huu ni wa kawaida kwa mstari CD, kwa hivyo kama vekta ya mwelekeo, tunatafuta mstari ambao ni orthogonal hadi overrightarrowCD overrightarrow D - mshale wa kulia zaidiC.Hebu katikati ya duara iwe asili, hivyoarrowoverrightA mshale uliopitilizaB arrow overrightarrowC arrow overrightarrowD.Kishaarrow overrightarrowC overrightarrowD cdot overrightarrowD - overrightarrowC overrightarrowD cdot overrightarrowD - overrightarrowC cdot overrightarrowC overrightarrowD2 - overrightarrowC2 0,hivyo vekta overrightarrowC overrightarrowD na overrightarrowD - overrightarrowC ni orthogonal. Kwa hivyo, mstari unaopita katikati ya overlineAB unaweza kuainishwa naoverrightarrowP frac12 overrightarrowA frac12 overrightarrowB t overrightarrowC overrightarrowD .Tukichukua t frac12, basi tutapokeaoverrightarrowP frac12 overrightarrowA frac12 overrightarrowB frac12 overrightarrowC frac12 overrightarrowD.Usemi huu ni linganifu kwa heshima na nukta zote nne, kwa hivyo nukta inayolingana P iko kwenye mistari yote sita. Kwa hivyo, a b c d frac12, na a b c d 2. 