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Area of the Triangle

Source: 1986 National High School Mathematics League, Exam Two, Problem 2

February 24, 2020
geometrycircumcircle

Problem Statement

In acute triangle ABCABC, D∈BC,E∈CA,F∈ABD\in BC,E\in CA,F\in AB. Prove that the necessary and sufficient condition of AD,BE,CFAD,BE,CF are heights of △ABC\triangle ABC is that S=R2(EF+FD+DE)S=\frac{R}{2}(EF+FD+DE). Note: SS is the area of △ABC\triangle ABC, RR is the circumradius of △ABC\triangle ABC.
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