Thursday, October 8, 2015

Geometry Problem 1151: Circle, Diameter, Chord, Perpendiculars, 90 Degrees, Metric Relations

Geometry Problem. Post your solution in the comments box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Click the diagram below for more details.

Online Math: Geometry Problem 1151: Circle, Diameter, Chord, Perpendicular, 90 Degrees, Metric Relations.

3 comments:

  1. Let the radius be R and let OE = s
    Let AG = a. GF = b and GH = c

    From similar Tr.s

    (R+s)(R-s) = CE^2 = ED^2. (1)

    (R+s)(2R) = a(a+b). (2)
    (R+s)^2 = a(a-c). (3)

    (2) - (3)
    (R+s)(R-s) = a(a+b) - a(a-c) = a(b+c) which from (1) equals CE ^2 which is the required result.

    Sumith Peiris
    Moratuwa
    Sri Lanka

    ReplyDelete
  2. Since BEGF concyclic, thus

    CE² = AE×EB
    = AE×AB - AE²
    = AG×AF - AG×AH
    = AG×HF

    ReplyDelete
  3. For (1) I've used similar Tr.s AED & BED, for (2) Tr.s ABF and AEG and for (3) Tr.s AEH & AEG

    ReplyDelete