## Algebraic Surfaces by Oscar Zariski

By Oscar Zariski

The most objective of this booklet is to give a totally algebraic method of the Enriques¿ class of gentle projective surfaces outlined over an algebraically closed box of arbitrary attribute. This algebraic method is likely one of the novelties of this publication one of the different smooth textbooks dedicated to this topic. chapters on floor singularities also are incorporated. The e-book could be invaluable as a textbook for a graduate path on surfaces, for researchers or graduate scholars in algebraic geometry, in addition to these mathematicians operating in algebraic geometry or comparable fields"

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FIGURE 1-34 Chapter 1 ELEMENTARY EUCLIDEAN GEOMETRY REVISITED 23 4. GIVEN: BE and AD are altitudes (intersecting at H) of AABC, while F, G, and K are midpoints of AH, ABy and RC, respectively (see Figure 1-35). FGK is a right angle. B A 5. A line PQ , parallel to base BC of AAPC, intersects AB and AC at points P and Q, respectively (see Figure 1-36). The circle passing through P and tangent to AC at Q intersects AB again at point P. Prove that points P, Q, C, and P are concyclic. As you proceed through the rest of this book, you may want to work with additional exercises.

This time, bisect LAGH and ACHG. These bisectors meet at point K. Once again, we notice that the required angle bisector (that of /-P) must contain point K. Because this required angle bisector must contain both J and iC, these two points determine our desired line, which is the location of the wire to be installed. • This solution relies heavily on the notion that the angle bisectors of a trian gle are concurrent. As we have said, the topic of concurrency in a triangle deserves more attention than it usually gets in the elementary geometry course.

Recall from elementary geometry that there are many “centers” of a triangle. Some examples are: centroid—the center of gravity of the triangle, determined by the intersec tion of the medians; orthocenter—the point of intersection of the altitudes of the triangle; incenter— the center of the inscribed circle of the triangle, determined by the intersection of the angle bisectors of the triangle; Chapter 2 CONCURRENCY of LINES in a TRIANGLE 27 ■ circumcenter—the center of the circumscribed circle (or circumcircle), determined by the intersection of the perpendicular bisectors of the sides of the triangle.