# Prove the equality of two line segments with circles

There are two circles. There are also two common "outside tangents" (they are marked red in the diagram) and one "inside tangent" which is marked blue. (I am not a native English speaker, I don't know how to call the outside and the inside tangents correctly) The inside tangent intersects the circles in points A and B and intersects two outside tangents in points A1 and B1. How can I prove that the line segment AA1 equals BB1?

• The inside tangent intersects the circles in points A1 and B1 Your diagram does not agree with this description ... – Antoine Apr 17 '17 at 20:22
• This has been fixed. – student28 Apr 17 '17 at 20:24
• What else do you know about the two circles that you didn't mention? How do you know that $B_1B=5.67$ and $AA_1=5.67$? – Mercy King Apr 17 '17 at 21:15
• Nothing else. The numbers are just to show that the problem is correct. (measured in GeoGebra). I guess we must find some congruent triangles here. @MercyKing – student28 Apr 17 '17 at 21:16
• It is equivalent to show that $AB_1=BA_1$. – Jean Marie Apr 17 '17 at 21:22

Let $E$ be the point of tangency of the lower tangent and the circle on the left. Then $AA_1 = AE$.
Let $F$ be the point of tangency of the lower tangent and the circle on the right.
$AA_1 = EF - A_1B$. $\ BB_1 = E_1F_1 (upper circle) - AB_1$.
Now $AA_1 = A_1B_1 - AB_1$; $BB_1 = A_1B_1 - A_1B$.
From these equations we get that $AB_1 = A_1B$.
• [+1] I have tried to improve a little your presentation. In particular, use underscores between a letter and its index, example A underscore 1 makes $A_1$. Besides, something is missing in your argumentation: you should mention the symmetry of the figure with respect to axis $A_2C$. – Jean Marie Apr 18 '17 at 10:03
• @student28 How is $AA_1 = EF - A_1B$?. – Narasimham Apr 18 '17 at 12:51