Contact of Two Circles

Here we will prove that two circles with centres X and Y touch externally at T. A straight line is drawn through T to cut the circles at M and N. Proved that XM is parallel to YN.

Contact of Two Circles


Given: Two circles with centres X and Y touch externally at T. A straight line is drawn through T to cut the circles at M and N.

To prove: XM YN.

Construction: Join T to X and Y.




1. In ∆XMT, ∠XMT = ∠XTM

1. XM = XT, being radii.

2. In ∆YNT, ∠YNT = ∠YTN

2. YN = YT, being radii.

3. XTY is a straight line.

3. The point of contact of two circles lies on the straight line joining their centres.

4. ∠XTM = ∠YTN

4. Vertically opposite angles.

5. ∠XMT = ∠YNT

5. From statements 1, 2 and 4.

6. XM ∥ YN. (Proved)

6. Alternate angles are equal, using statement 5.

10th Grade Math

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