Yagci

Yagci's problem

In a letter to the Hyacinthos geometry group, Mostafa Yagci proposed to prove or disprove that the triangle DEF, defined through the projections D, E, F of the feet H, G, I of the altitudes of the triangle is a triangle similar to ABC. Indeed it is always similar to ABC. This is discussed below, together with some additional remarks.

DEF is pivotal triangle of ABC around the first Brocard point J of ABC. There is another similar triangle resulting from the other orientation of the triangle and related to the second Brocard point of ABC. Anyway, to see that DEF is pivotal one does an easy calculation of the ratios (|BE|:|CD|:|AF|) and shows that they are equal to ((c/b):(a/c):(b/a)). Thus, applying the well known criteria (see BrocardPivot.html ), DEF is pivotal around the Brocard point J and similar to ABC. J is also the first Brocard point of triangle KLM.
The circles {BEF}, {CED} and {ADF} are tangent to the sides ED at E, DF at D and FE at F respectively. Joining an arbitrary point N of the circle {CED} with the vertices D, E and intersecting with the other two circles, we create external pivotal triangles NOP. Point J is the first Brocard point of all these triangles. Their symmedian point S moves on the Brocard circle of DEF, as N varies on {CED}.

There is a particular position of N, where the sides of NOP are orthogonal to those of DEF. Then S coincides with the circumcenter Q of DEF. The corresponding configuration is then identical to the one discussed in Brocard2.html .

http://users.math.uoc.gr/~pamfilos/