Sunday, August 5, 2018

Geometry Problem 1374: Isosceles Triangle, Exterior Cevian, Incircle, Excircle, Tangency Points, Parallel Lines

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

Details: Click on the figure below.

Geometry Problem 1374: Isosceles Triangle, Exterior Cevian, Incircle, Excircle, Tangency Points, Parallel Lines.

1 comment:

  1. Let the altitude BKL of Tr. ABC meet AC at L and the parallel to AC thro E at K.

    Let BL = h, AB = BC = a, BF = a - b, CH = c, AL = LC = d

    Let < FBE = < FKE = x, BFEK being cyclic
    Let < KHL = y

    As per my solution to Problem 1373, BK = h - r1 = r2 = GH
    So BKHG is a parallelogram and so KH //BG

    Now tan x = r1/(a-b) and tan y = r1/(c + d)

    But since a - b = c + d again as per problem 1373, tan x = tan y
    Hence x = y
    So FKH must be collinear since EK // AH

    So finally FKH // BG

    Sumith Peiris
    Moratuwa
    Sri Lanka

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