Saturday, April 4, 2015

Problem 1107: Triangle, Circumcircle, Circle, Radius, Tangent, Chord, Congruence

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

Click the diagram below to enlarge it.

Online Math: Geometry Problem 1107: Triangle, Circumcircle, Circle, Radius, Tangent, Chord, Congruence.



    Let circle D cut AB at E and F
    Since DC^2=DA.DB => Circle D is Apollonius circle of AB
    And PE and CE are bisectors of angles APB and ACB
    So PA/PB=EA/EB=CA/CB (1)
    Triangles PBC similar to PC1B1 => B1C1= (PC1/PB). BC…..(2)
    Triangles PAC similar to PC1A1=> C1A1= (PC1/PA). CA …..(3)
    Divide (2) to (3) and using (1) we get B1C1= C1A1

  2. Let AB cut circle D at Q.
    Let < BCD = p, <BPD = q, < QPC = r and < QCP = s

    Then <CAD = p and < PAD = q since DP (=DC) is a tangent to circle APB at P.
    Further < PDA = 2s and < CDA = 2r so that
    < PQD = < QPD = 90-s …..(1) and
    < QCD = CQD = 90-r…….(2)

    From (2) < ACQ = 90-p-r …..(3)
    From (1) < APQ = 90-q-s ……(4)

    Now < ABB1 = 2s+q & …..(5)
    And < CBA = 2r+p ……(6)

    Further < PCB = 90-p-r-s from which
    < A1B1C1 = 90-p-q-r-s …..(7) since < PCB = < PB1C1 and < BB1A = BAA1 = q

    Also < B1A1C1 = < PA1C1 - <PA1B1 = < ACC1 - < ABB1 = (90-p-r+s) – (2s+q) =

    From (7) and (8), < A1B1C1 = < B1A1C1 and therefore C1A1 = C1B1

    Sumith Peiris
    Sri Lanka

  3. Dear Antonio Gutierrez,
    I noticed that you deleted my solution in Russian to this problem. Will it be acceptable in the future
    if I post side by side English-Russian translations of my solutions so that Russian readers can understand them?

    1. Thanks Ivan,
      Problem 1107: Could you post your solution in Russian again? I will publish it

      Thanks again

    2. Thank you so much! It means the world to me if I can spread this knowledge in my native language.

  4. Here is my solution, though rushed you should be able to deduct from my reasoning, and if necessary I may repost the solution in case the wording is difficult to understand.

  5. Dear Barth
    Referring to line 3 of your solution " from triangle A1B1C angle A1 is 90 degrees... "

    Per the problem statement , P is any point on circle center D, radius DC. It may exist a particular point P so that C, O and B1 are collinear, but in general these 3 points are not collinear and angle A1 in triangle A1B1C is not equal 90 degrees. please provide explanation.

    Peter Tran