Two circles intersect at points A and B. Through points A and B, two straight lines are drawn that intersect one circle at points M and N, and the other circle at points K and L, and point A lies between points M and K, and point B between points N and L. Prove that lines MN and KL are parallel.
The quadrangles AMНB and AKLM are inscribed in the circle, then in these quadrangles the sum of opposite angles is 180.
Angle AMН + ABH = MAB + BHM = 180, BAK + BLK = AKL + ABL = 180.
The angle MAB and BAK are adjacent angles, then MAB = 180 – BAK.
180 – BAK + BHM = 180.
Then BАК = BНМ.
Then BHM = 180 – BLK.
Angle BLC adjacent to the angle КLС, BLC = 180 – КLС.
Then BHM = 180 – 180 + КLС.
ВНM = KLС.
The angles BНМ and KLС are the corresponding angles at the intersection of straight lines МН and KL secant НL, and since they are equal, МН and KL are parallel, which was required to be proved.
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