By Alfred S. Posamentier
Advanced Euclidean Geometry provides an intensive overview of the necessities of high college geometry after which expands these recommendations to complicated Euclidean geometry, to provide lecturers extra self assurance in guiding scholar explorations and questions.
The textual content includes countless numbers of illustrations created within the Geometer's Sketchpad Dynamic Geometry® software program. it's packaged with a CD-ROM containing over a hundred interactive sketches utilizing Sketchpad™ (assumes that the consumer has entry to the program).
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Extra info for Advanced Euclidean Geometry
As we have mentioned, this theorem, which today bears Menelaus’s name, did not become popular until it was rediscovered by Giovanni Ceva as a part of his work in 1678. 1 (Menelaus^S theorem) The threepoints P, Q, 1 - ЛГАЛ 11. TlQ and BC, respectively, of /лАВС are collinear if andonly i f -----• ^ ^ ^ QB ^^ and R onthesidesAC,AB, BR CP — • — = —1. RC PA Like Ceva’s theorem, Menelaus’s theorem is an equivalence and therefore requires proofs for each of the two statements (converses of each other) that comprise the entire^theorem.
TlQ and BC, respectively, of /лАВС are collinear if andonly i f -----• ^ ^ ^ QB ^^ and R onthesidesAC,AB, BR CP — • — = —1. RC PA Like Ceva’s theorem, Menelaus’s theorem is an equivalence and therefore requires proofs for each of the two statements (converses of each other) that comprise the entire^theorem. We will first prove that if the three points P, Q, and R on the sides AC, AB, and BC,, respectively, of AABC are collinear, then AQ BR CP TT;: * ~ L We offer two proofs of this part of Menelaus’s theorem.
Because the bisector (AL) of an interior angle of a triangle partitions the opposite side proportionally to the remaining two sides of the triangle: (I) LC ~ AC An exterior angle bisector partitions the side that it intersects proportionally to the remaining sides of the triangle. This property produces the following proportions: < > CM For B M : MA BC AB (II) i > AN For CN : NB AC BC (III) — By multiplying (I), (II), and (III), we get: ^ CM AN AB BC AC L C ' m a ’ N B ~ A C ' A b ' BC 34 ADVANCED EUCLIDEAN GEOMETRY N INTERACTIVE 2-9 Drag vertices A B, and CXo change the shape of the triangle and see that the indicated bisectors of the angles (interior and exterior) always meet at one point.
Advanced Euclidean Geometry by Alfred S. Posamentier