# 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.

Solution:

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.

Proof:

 Statement Reason 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.

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