Question

Two circles intersect at points M and N. Through M, the diameters MA and MB of the two circles are drawn. Show that A, N and B are collinear.

Answer 1

Given: 

Two circles intersect at points M and N. MA and MB are diameters of the two circles passing through M.

Step 1: Recognise diameters subtend right angles to the circle.

Since MA and MB are diameters of the respective circles, both the angles ∠MNA and ∠MNB are right angles (90°).

Because the angle subtended by a diameter on any point of the circle is 90°.

Step 2: Identify right angles in triangles.

Consider triangle ANB.

Since ∠MNA = 90°, point N lies on the circle with diameter MA.

Similarly, since ∠MNB = 90°, point N lies on the circle with diameter MB.

Step 3: At point N:

NA ⊥ NM

NB ⊥ NM

Therefore, NA and NB are the same straight line (both perpendicular to NM).

Thus A, N, B are collinear.