Problem 3
ABCD is a quadrilateral such that a circle can be insribed in it. Four circles are drawn inside ABCD, such that they touch each other and also the sides of the quadrilateral.
Let EF and GH be the common tangents of these four circles (that are not the sides of ABCD). Then, prove that:
EF = GH
Here’s a simulation that shows what needs to be proven.
You can drag the points A, H, and D, and observe the lengths of EF and GH.
Solution
I won’t provide the full solution here, but only give you some tools and hints that’ll help you solve the problem yourself.
So, tangents. Lot’s of them!
There’s only one property about tangents that you need to know to be able to solve this problem.
That is, the tangents drawn to a circle from an external point are equal in length. Here’s a little simulation, where you can see this property in action.
You can drag the point P and observe the lengths of PA and PB. Are they always equal?
Well, that’s all that you need to know here. I suggest you proceed on your own from this point. Good luck!
Were you able to solve it using the above hints? Or, did you solve using any other method? Or, do you need some more help? You can reach out at the Facebook page.
Problem Source: GoGeometry
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