Touching Circles and Concurrency

The applet below illustrates the following problem that has been sent by Ionut Onisor, Bucharest, Romania:

Let AH_a, BH_b and CH_c be the altitudes of ΔABC. Draw a circle through H_b and H_c tangent to circumcircle. Let A' be the point of tangency and B' and C' similarly defined. Prove that lines AA', BB' and CC' have a common point.

As you can check with the applet, the statement requires qualifications: first, it may only be true for acute triangles. Second, there may be two circles through H_b and H_c tangent to circumcircle. The statement only holds when we pick the circle such that triangles AH_bH_c and A'H_bH_c have different orientation; in other words, when line H_bH_c separates A and A'.

In addition, the lines A'H_a, B'H_b, C'H_c are also concurrent and the point of concurrency appears to lie on the Euler line of ΔABC.

Solution

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Copyright © 1996-2012 Alexander Bogomolny

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.

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... to be continued ...

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Copyright © 1996-2012 Alexander Bogomolny