Question

A, B, C, D are 4 non collinear points in a plane such that $\angle$ ACB $=\angle$ ADB, then how many circle(s) can be drawn passing through all 4 points.

• A. True
• B. False
• C. 2
• D. 3

Answer 1

The correct option is B

False

Let us draw the points A, B, C, D such that $\angle$ACB $=\angle$ADB.

Now, draw a circle through A, B, C. Let us assume that the circle does not pass through the point D but intersects the extension of line segment CD at $D^{{}^{\prime }}$.

Since, angles subtended by an arc in a segment are equal, $\angle$ ACB $=\angle$A$D^{{}^{\prime }}$B. It is given that $\angle$ACB $=\angle$ADB. Thus, for the angles to be equal, D and $D^{{}^{\prime }}$ should coincide. Thus, our assumption that the circle does not pass through D is wrong.

Therefore, only one circle can be drawn through these 4 points.