We strengthen a result of Lehmer, obtaining a new necessary condition for the roots of a complex polynomial to have equal modulus. From this we derive the famous theorem of Feuerbach, as well as the less well-known theorems of Euler and Guinand on the tritangent centers of a triangle. The latter theorems constrain the possible locations of the incenter and excenters subject to fixed locations for the circumcenter and orthocenter.
We give a simple proof of Euler's remarkable theorem that for a non- degenerate triangle, the set of points eligible to be the incenter is precisely the orthocentroidal disc, punctured at the nine-point center. The problem is handled algebraically with complex coordinates. In particular, we show how the vertices of the triangle may be determined from the roots of a complex cubic whose co- efficients are functions of the classical centers.
The economic performance of Sub-Saharan Africa (SSA) over the past two decades has been disappointing. The initial explanation for this poor performance focussed on deficient macroeconomic policies. However a number of SSA countries have undertaken and sustained a reasonable measure of macroeconomic reform and still see no significant expansion in the growth of manufactured exports. This suggests that a stable, credible, and consistent macroeconomic environment may be a necessary but not sufficient condition for export growth. This paper reviews the literature dealing with other constraints, including poor infrastructure, lack of appropriate human capital, a failure to develop a variety of export support services, and the impact of an unresponsive civil service. We conclude that while there are clearly numerous structural barriers to exports in SSA, no systematic analysis has been undertaken that would assist policy-makers in determining which constraints are most important. Joseph J. Stern, Institute Fellow at HIID, currently resides in Indonesia where he serves as Project Coordinator for HIID Indonesia projects. He was previously resident in Malaysia, Ghana, and Pakistan. Mary Kay Gugerty is a doctoral student in the Program in Political Economy and Government at Harvard University.
Five patients with electrocardiographic abnormalities simulating early myocardial infarction after scorpion sting are reported. Pulmonary edema and congestive heart failure accompanied these electrocardiographic changes in 3 patients. Serial recordings revealed an evolution to normalcy.
Ebstein's anomaly has been reported in a family affecting a brother and a sister. The rest of the family members, including the nonidentical twin brother of the second patient, were free of symptoms. The clinical and hemodynamic data indicated Ebstein's anomaly in both. This was confirmed on necropsy of the first patient and at operation in the second, in whom a prosthetic valve replacing the tricuspid was inserted at the age of 13 years. This patient also died eight months following operation. Despite this outcome and from the experience of others, it is our belief that certain cases of Ebstein's anomaly may be amenable to surgical treatment. The available surgical procedures and the indications for operation are briefly discussed.