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geometry puzzle (18) |  |
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Please look at the picture below:
The question is to calculate angle x.
ΔABC is isosceles, AC = BC, because LB = LA = 70.
Point D is the center of the circle through C and E.
This circle intersects BC in point H.
Draw lines CE and EH.
Draw perpendicular bisector of AB.
Draw point G on CF such, that HG is perpendicular to CF.
Now look at ΔADC.
LADC = 180 - 20 - 40 = 120
LADB = 180 - 120 = 60.
DE = DH so LDEH = LHED = (180 - 60)/2 = 60.
G is on the circle because LCGH = 90.
LHEG = LBCF = 20 because both span arc GH.
x = LDEB = 20 + 60 = 80.
This leaves the question: is point G on line BE?
LAEB = 180 - 60 - 20 = 100.
LABE = 180 - 100 - 50 = 30.
LFGB = 90 - 30 = 60.
LCGE = 120/2 = 60 because it spans arc CE.
So point G is on line BE.
