Ptolemy theorem will not be given in the senior high school entrance examination, because Ptolemy theorem is not involved in the scoring standard of the senior high school entrance examination mathematics exam.
Ptolemy theorem is not recommended to be used casually in the senior high school entrance examination. If you want to use it, you need to prove it first. Ptolemy theorem means that the sum of the products of two opposite sides of a circle inscribed with a convex quadrilateral is equal to the product of two diagonals.
Theorem statement: In the inscribed quadrilateral of a circle, the area of a rectangle surrounded by two diagonal lines is equal to the sum of the area of a rectangle surrounded by one set of opposite sides and the area of a rectangle surrounded by another set of opposite sides. From this theorem, the sum and difference formulas of sine and cosine and a series of trigonometric identities can be derived. Ptolemy theorem is essentially about the basic properties of circles.
Verification and deduction of Ptolemy theorem;
In any convex quadrilateral ABCD, let △ABE make ∠BAE=∠CAD ∠ABE=∠ ACD and connect DE, then △ABE∽△ACD. So BE/CD=AB/AC, that is, be AC = AB CD (1) gets AD/AC=AE/AB from △ABE∽△ACD, and △ BAC = ∠ EAD, so △ ABC ∽△ AED.BC/ED=AC/AD, is ED.
(1)+(2), AC (be+ed) = ab CD+ad BC, and because BE+ED≥BD. (Only when the quadrilateral ABCD is inscribed with a circle, the equal sign holds, that is, Ptolemy's theorem).
Ptolemy theorem is an important theorem in geometry, which can be used to solve some problems related to circles, such as proving that quadrangles are rectangles. Although it will not be used in the senior high school entrance examination, learning Ptolemy theorem can deepen the understanding of geometry and improve the level of mathematics.
Studying Ptolemy's theorem can also lay a foundation for the future college entrance examination or other math competitions, because in these exams, the content of geometry will be deeper and more complicated, and Ptolemy's theorem will be more widely used.