SOLUTION: Two circles intersect at A and B.From B two diameters BC and BD are drawn which cut the circles at C and D respectively.Prove that CAD is a straight line.

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Question 1068288: Two circles intersect at A and B.From B two diameters BC and BD are drawn which cut the circles at C and D respectively.Prove that CAD is a straight line.
Found 2 solutions by KMST, ikleyn:
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
A and B are points on both circles.
C is on one circle,
where BC is a diameter.
Since BC is a diameter, angle BAC is an inscribed angle
intercepting an arc that is half of that circle.
The measure of BAC is half of the intercepted arc,
or 180%5Eo%2F2=90%5Eo .
The same reasoning with diameter BD on the other circle
proves that angle BAD measures 90%5Eo .
Since BAC and BAD both measure 90%5Eo ,
their sum, angle CAD measures
90%5Eo%2B90%5Eo=180%5Eo ,
meaning that rays AC and AD are two "halves" of the same line CD.

Answer by ikleyn(52866) About Me  (Show Source):
You can put this solution on YOUR website!
.
0.  Make a sketch to follow my arguments.


1.  Consider the angle CAB.

    This angle is right angle, since it leans the diameter.


2.  Consider the angle DAB.

    This angle is right angle, since it leans the diameter.


3.  Since both angles CAB and DAB are right angles, the rays AC and AD make one straight line.

Proved and solved.