In his high school final examination (Aarau, Switzerland, 1896), young Albert Einstein

Chapter 7, Problem 41

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In his high school final examination (Aarau, Switzerland, 1896), young Albert Einstein (1879-1955) was given the following problem: In a triangle ABC, let P be the center of the inscribed circle. We are told that AP = 1, BP = 2, and CP = 3. Find the radius p of the inscribed circle. Einstein worked through this problem as follows: (!)= < n ( ^ =in ( I ) = 3 ,.For every triangle the following equation holds: !i"2(I)+si"2(f)+sin2(l)+ 2 sn ( i ) s", ( f ) sin( i ) = 1'In our caseNow let14/02 + 12p 3 - 1 = 0.P = - JC At this point we interrupt Einsteins work and ask you to finish the job. [Hint: Exercise 40 is helpful. Find the exact solution (in terms of trigonometric and inverse trigonometric functions), and give a numerical approximation as well.] (By the way, Einstein, who was allowed to use a logarithm table, solved the problem correctly.) (iSource: The Collected Papers ofA lbert Einstein, Vol. 1, Princeton University Press, 1987.)