Wednesday, October 17, 2012

Problem 815: Triangle, Incircle, Semicircles, Arbelos, Area

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the complete problem 815.

Online Geometry Problem 815: Triangle, Incircle, Semicircles, Arbelos, Area

3 comments:

  1. Let BC=a, AC=b, AB=c, s=1/2(a+b+c).

    Then
    AD=AF=s−a, BD=BE=s−b, CE=CF=s−c

    Sa
    = 1/8 π[a^2 − (s−b)^2 − (s−c)^2]
    = 1/8 π[a^2 − 1/4 (a+c−b)^2 − 1/4 (a+b−c)^2]
    = 1/8 π[1/2 a^2 − 1/2 (b−c)^2]
    = 1/16 π[(a+b−c)(a−b+c)]
    = 1/4 π (s−b)(s−c)

    1/Sa = 4/π 1/[(s−b)(s−c)]
    1/Sb = 4/π 1/[(s−a)(s−c)]
    1/Sc = 4/π 1/[(s−a)(s−b)]

    1/Sa + 1/Sb + 1/Sc
    = 4/π [1/[(s−b)(s−c)]+1/[(s−a)(s−c)]+1/[(s−a)(s−b)]]
    = 4/π s/[(s−a)(s−b)(s−c)]
    = 4/S

    ReplyDelete
  2. Let AB = c, BC = a, CA = b, and s = (a+b+c)/2
    Then BD = BE = (s-b), EC = FC = (s-c), FA = AD = (s-a)
    The area of arbelo
    Sa = π((BE/2)(EC/2)) = π(s-b)(s-c)/4
    Similarly,
    Sb = π(s-a)(s-c)/4
    Sc = π(s-a)(s-b)/4

    1/Sa + 1/Sb + 1/Sc
    = 4[(s-a)+(s-b)+(s-c)] / π[(s-a)(s-b)(s-c)]
    = 4s/π[(s-a)(s-b)(s-c)]
    = {4s/π[(s-a)(s-b)(s-c)]}
    = (1/π)(4s/[(s-a)(s-b)(s-c)])
    = (1/π)(s/Δ)^2, where Δ is the area of triangle ABC
    = (1/π)(1/r)^2, where r is the inradius
    = 1/S

    ReplyDelete
    Replies
    1. *Typo :

      For the last 3 lines of my proof:
      = 4(1/π)(s/Δ)^2, where Δ is the area of triangle ABC
      = 4(1/π)(1/r)^2, where r is the inradius
      = 4/S

      Delete