Wednesday, September 4, 2024

Geometry Problem 1580: 60-Degree Triangle Challenge: Find AD with Excenter and Circumcircle Clues!

Challenging Geometry Puzzle: Problem 1580. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Illustration of Geometry Problem 1580

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2 comments:

  1. https://photos.app.goo.gl/fn3xiPK7gUCenPeb6

    Define points I, M per sketch where I is incenter of ABC
    Note that B, D,C, I are concyclic
    And ^(DBC)=^(DIC)= 60 and ^(IDC)=30
    Angle AEC= ½ arc(ABC)= 120 => ^(DAE)=180-120-30=30
    Si triangle ADE is isosceles and AD= 2.DE.cos(30)= 10.sqrt(3)

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  2. Let the tangency points on the Excircle D be U,V,W for AB, BC, CA respectively

    So < BCD = < DCW = @ (say)
    Hence < BCW = 2@ and < BAC = 2@ - 60

    Now AD bisects < BAC
    So < DAC = 1/2. (2@ - 60) = @ - 30
    And < ADC = @ - (@ - 30) = 30
    So AED is a 30-30-120 Isosceles Triangle

    Hence AD = 2. DE cos 30 = 10 V3

    Sumith Peiris
    Moratuwa
    Sri Lanka

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