Two Intersecting Circles: What is this about?
A Mathematical Droodle

 

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Explanation

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Copyright © 1996-2017 Alexander Bogomolny

The applet illustrates a problem suggested by Bui Quang Tuan:

 

Let two circles C(P) and C(Q) intersect in points C and D. A line through C intersects the second time C(P) at A and C(Q) at B. Let O be the midpoint of PQ. Then the circle C(O) with center O through C and D meets AB at the midpoint T.

 

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

The fact is practically algebraic.

Let X, Y, Z be the midpoints of AB, AC, and BC, respectively. Introduce a = AY = YC, c = TX = XC, and b = CZ = ZB. Any of the numbers a, b, c may be negative if the direction of the corresponding segments does not agree with that of AB.

For T to be the midpoint of AB we should have 2a + 2c = 2b - 2c, or a + c = b - c. But this is exactly the condition for X to be the midpoint of YZ, which it is (because PO = OQ.)

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Copyright © 1996-2017 Alexander Bogomolny

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