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Question
In the figure, AB is common chord of the two circles. If AC and AD are diameters; prove that D, B and C are in a straight line. O1 and O2 are the centers of two circles.
Solution
Given: Two circles with centre O1 and O2 intersect each other at A and B. AC and AD are the diameters of the circles.
To prove: D, B, C are in the same straight line.
Construction: Join AB.
Proof:
AO1C is diameter.
∠ABC = 90° ...(Angle in a semi-circle)
Similarly, ∠ABD = 90°,
Adding, we get:
∠ABC + ∠ABD = 90° + 90° = 180°
DBC is a straight line or D, B, C are in the same line.
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